Rev. Math. Phys. vol. 15, No. 5 (2003) 447-489
ABSTRACT: Given a $W^*$-algebra $\fM$ with a $W^*$-dynamics $\tau$, we prove the existence of the perturbed $W^*$-dynamics for a large class of unbounded perturbations. We compute its Liouvillean. If $\tau$ has a $\beta$-KMS state, and the perturbation satisfies some mild assumptions related to the Golden-Thompson inequality, we prove the existence of a $\beta$-KMS state for the perturbed $W^*$-dynamics. These results extend the well known constructions due to Araki valid for bounded perturbations.
Ann. H. Poincare, vol 4, 1001-1013 (2003)
ABSTRACT: We derive the first six coefficients of the heat kernel expansion for the electromagnetic field in a cavity by relating it to the expansion for the Laplace operator acting on forms. As an application we verify that the electromagnetic Casimir energy is finite.
Commun. Math. Phys. 243, 241-260 (2003) Springer-Verlag Heidelberg ISSN: 0010-3616 (Paper) 1432-0916 (Online) DOI: 10.1007/s00220-003-0958-6
ABSTRACT: Starting from a formal Hamiltonian as found in the physics literature -- omitting photons -- we define a renormalized Hamiltonian through charge and mass renormalization. We show that the restriction to the one-electron subspace is well-defined. Our construction is non-perturbative and does not use a cut-off.
Ann. H. Poincaré 4 (2003) 713 - 738
ABSTRACT: Quadratic bosonic Hamiltonians with a linear perturbation are studied. Depending on the infrared and ultraviolet behavior of the perturbation, their properties are described from the point of view of spectral and scattering theory.
Ann. H. Poincaré 4 (2003) 739 - 793
ABSTRACT: We study ergodic properties of Pauli-Fierz systems---$W^*$-dynamical systems often used to describe the interaction of a small quantum system with a bosonic free field at temperature $T\geq0$. We prove that, for a small coupling constant uniform as the positive temperature $T\downarrow 0$, a large class of Pauli-Fierz systems has the property of return to equilibrium. Most of our arguments are general and deal with mathematical theory of Pauli-Fierz systems for an arbitrary density of bosonic field.
J. Stat. Phys. vol.166, 425-473 (2004)
ABSTRACT: This paper is about adiabatic transport in quantum pumps. The notion of ``energy shift'', a self-adjoint operator dual to the Wigner time delay, plays a role in our approach: It determines the current, the dissipation, the noise and the entropy currents in quantum pumps. We discuss the geometric and topological content of adiabatic transport and show that the mechanism of Thouless and Niu for quantized transport via Chern numbers cannot be realized in quantum pumps where Chern numbers necessarily vanish.
Ann. Henri Poincare 5 (2004), no. 4, 671-741
ABSTRACT: The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. A new Lieb-Thirring type inequality on the sum of the negative eigenvalues is presented. The main feature compared to earlier results is that in the large field regime the present estimate grows with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain an optimal upper bound on the pointwise density of zero energy eigenfunctions of the Dirac operator. The main technical tools are: (i) a new localization scheme for the square of the resolvent of a general class of second order elliptic operators; (ii) a geometric construction of a Dirac operator with a constant magnetic field that approximates the original Dirac operator in a tubular neighborhood of a fixed field line. The errors may depend on the regularity of the magnetic field but they are uniform in the field strength.
Jour. Stat. Phys., 116 (1-4): 475-506, August 2004
ABSTRACT: The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. Bounds on the sum of the negative eigenvalues are called magnetic Lieb-Thirring (MLT) inequalities. The purpose of this paper is twofold. First, we prove a new MLT inequality in a simple way. Second, we give a short summary of our recent proof of a more refined MLT inequality \cite{ES-IV} and we explain the differences between the two results and methods. The main feature of both estimates, compared to earlier results, is that in the large field regime they grow with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain optimal upper bounds on the pointwise density of zero energy eigenfunctions of the Dirac operator.
Ann. Henri Poincare 4, 1101-1136, (2003)
ABSTRACT: We examine the binding conditions for atoms in non-relativistic QED, and prove that removing one electron from an atom requires a positive energy. As an application, we establish the existence of a ground state for the Helium atom.
Henri Poincare 5, 1137 - 1157 (2004)
ABSTRACT: We consider an external potential, $-\lambda \phi$, due to one or more nuclei. Following the Dirac picture such a potential polarizes the vacuum. The polarization density as derived in physics literature, after a well known renormalization procedure, depends decisively on the strength of $\lambda$. For small $\lambda$, more precisely as long as the lowest eigenvalue, $e_1(\lambda)$, of the corresponding Dirac operator stays in the gap of the essential spectrum, the integral over the density vanishes. In other words the vacuum stays neutral. But as soon as $e_1(\lambda)$ dives into the lower continuum the vacuum gets spontaneously charged with charge $ 2e$. Global charge conservation implies that two positrons were emitted out of the vacuum, this is, a large enough external potential can produce electron-positron pairs. We give a rigorous proof of that phenomenon.
Phys. Rev. Lett 92, 040201-1 (2004)
ABSTRACT: Deformations can induce rotation with zero angular momentum where dissipation is a natural ``cost function''. This gives rise to an optimization problem of finding the most effective rotation with zero angular momentum. For certain plastic and viscous media in two dimensions the optimal path is the orbit of a charged particle on a surface of constant negative curvature with magnetic field whose total flux is half a quantum unit.
Ann. Henri Poincare 5 (2004), no. 5, 871--914.
ABSTRACT: The goal of this paper is twofold. First, assuming strict convexity of the surface tension, we derive a stability property with respect to the Hausdorff distance of a coarse grained representation of the interface between the two pure phases of the Ising model. This improves the $\bbL^1$ description of phase segregation. Using this result and an additional assumption on mixing properties of the underlying FK measures, we are then able to extend to higher dimensions previous results by Martinelli on the spectral gap of the two-dimensional Glauber dynamics. Our assumptions can be easily verified for low enough temperatures and, presumably, hold true in the whole of the phase coexistence region.
Phys. Rev. Lett. 91, 150401-1-4 (2003)
ABSTRACT: Recent experimental and theoretical work has indicated conditions in which a trapped, low-density Bose gas ought to behave like the 1D delta-function Bose gas solved by Lieb and Liniger. Up to now the theoretical arguments have been based on variational - perturbative ideas or numerical investigations. There are 4 parameters: density, transverse and longitudinal dimensions, and scattering length. In this paper we explicate 5 parameter regions in which various types of 1D or 3D behavior occur in the ground state. Our treatment is based on a rigorous analysis of the many-body Schrodinger equation. This version of the article is a combination of the earlier 4-page arXiv version and the revised 4-page Physical Review Letters version.
Commun. Math. Phys. 244, 347-393 (2004)
ABSTRACT: Recent experimental and theoretical work has shown that there are conditions in which a trapped, low-density Bose gas behaves like the one-dimensional delta-function Bose gas solved years ago by Lieb and Liniger. This is an intrinsically quantum-mechanical phenomenon because it is not necessary to have a trap width that is the size of an atom -- as might have been supposed -- but it suffices merely to have a trap width such that the energy gap for motion in the transverse direction is large compared to the energy associated with the motion along the trap. Up to now the theoretical arguments have been based on variational - perturbative ideas or numerical investigations. In contrast, this paper gives a rigorous proof of the one-dimensional behavior as far as the ground state energy and particle density are concerned. There are four parameters involved: the particle number, $N$, transverse and longitudinal dimensions of the trap, $r$ and $L$, and the scattering length $a$ of the interaction potential. Our main result is that if $r/L\to 0$ and $N\to\infty$ the ground state energy and density can be obtained by minimizing a one-dimensional density functional involving the Lieb-Liniger energy density with coupling constant $\sim a/r^2$. This density functional simplifies in various limiting cases and we identify five asymptotic parameter regions altogether. Three of these, ºcorresponding to the weak coupling regime, can also be obtained as limits of a three-dimensional Gross-Pitaevskii theory. We also show that Bose-Einstein condensation in the ground state persists in a part of this regime. In the strong coupling regime the longitudinal motion of the particles is strongly correlated. The Gross-Pitaevskii description is not valid in this regime and new mathematical methods come into play.
ABSTRACT: We consider the grand canonical pressure for Coulombic matter with nuclear charges $\sim Z$ in a magnetic field $B$ and at nonzero temperature. We prove that its asymptotic limit as $Z\to\infty$ with $B/Z^3\to 0$ can be obtained by minimizing a Thomas-Fermi type pressure functional.
J. Stat. Phys, {\bf 116}, 523-546 (2004)
Ann. Henri Poincare 5 (2004) 523-577
ABSTRACT: We consider in this paper the scattering theory of infrared divergent massless Pauli-Fierz Hamiltonians.We show that the CCR representations obtained from the asymptotic field contain so-called {\em coherent sectors} describing an infinite number of asymptotically free bosons. We formulate some conjectures leading to mathematically well defined notion of {\em inclusive and non-inclusive scattering cross-sections} for Pauli-Fierz Hamiltonians. Finally we give a general description of the scattering theory of QFT models in the presence of coherent sectors for the asymptotic CCR representations.
Journ. Stat. Phys. 116 (2004) 411-423
ABSTRACT: e consider a general class of models consisting of a small quantum system $\cS$ interacting with a reservoir $\cR$. We compare three applications of 2nd order perturbation theory (the Fermi Golden Rule) to the study of such models:(1) the van Hove (weak coupling) limit for the dynamics reduced to ${\cal S}$; (2) the Fermi Golden Rule applied to the Liouvillean---an argument that was used in recent papers on the return to equilibrium; (3) the Fermi Golden Rule applied to the so-called C-Liouvillean. These three applications lead to three Level Shift Operators. As our main result, we prove that if the reservoir $\cR$ is thermal (if it satisfies the KMS condition), then the Level Shift Operator obtained in (1) (often called the Davies generator) and the Level Shift Operator constructed in (2) are connected by a similarity transformation. We also show that the Davies generator coincides with the Level Shift Operator obtained in (3) for a general $\cR$.
Multiscale methhods in Quantum Mechnaics, editors P. Blanchard and G.Dell'antonio, Birkhauser (2004).
ABSTRACT: I explain the two colored butterflies shown in figures 1 and 4, their thermodynamic significance and their duality.
J. Funct. Anal. 216 (2004), 1-21
ABSTRACT: We prove some sharp Hardy type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.
Arkiv for Matematik 42 (2004) 87-106.
ABSTRACT: We prove that the electronic densities of atomic and molecular eigenfunctions are real analytic in ${\mathbb R}^3$ away from the nuclei.
Accepted for publication in Canadian Journal of Mathematics.
ABSTRACT: In this paper, we consider the quantum version of the hamiltonian model describing friction introduced in [BDB]. This model consists of a particle which interacts with a bosonic reservoir representing a homogeneous medium through which the particle moves. We show that if the particle is confined, then the Hamiltonian admits a ground state if and only if a suitable infrared condition is satisfied. The latter is violated in the case of linear friction, but satisfied when the friction force is proportional to a higher power of the particle speed.
Published in J. Statist. Phys. vol. 116, no. 5, 1545-1578 (2004).
ABSTRACT: It has been known since the rigorous result by Angelescu and Corciovei [A-C] that there is no Bose-Einstein Condensation of the three dimensional Perfect Bose Gas (PBG) in a homogeneous magnetic field. The main result of the present paper is that the answer can become positive if the bosons are simultaneously embedded in a periodic external potential. We show that this is true for PBG, as well as for the Bose gas with a mean-field repulsive particle interaction.
Phys Rev Lett 91 issue 18 of Physical Review Letters with electronic identifier 186801
ABSTRACT: The magnetization and the de Haas-van Alphen oscillations of Bloch electrons are calculated near commensurate magnetic fluxes. Two phases that appear in the quantization of mixed systems--the Berry's phase and a phase first discovered by Wilkinson--play a key role in the theory.
Comm. in Math. Phys. 242, p. 501-529 (2003).
ABSTRACT: Let $H$ be a Schr\"odinger operator defined on $\mathbb R^d$ with smooth potential V, such that V is invaraint with respect to translation, i.e $V(x+n)=V(x)$ for all vectors $n$ in $\mathbb Z^d$. In addition assume that $V$ is invariant with respect to the reflections $T_j$ by the planes $x_j=0$. This is a periodic Schr\"odinger operator with additional reflection symmetries. We investigate the associated Floquet operators $H^q$ for $q\in [0,1]^d$. In particular we show that the associated lowest eigenvalues $\lambda_q$ are simple if all the $q_j\neq 1/2$ for each $j=1,\dots, d$.
Physics Today August 2003. p. 38-42
ABSTRACT: Topological quantum numbers account for the precise quantization that occurs in the integer Hall effect. In this theory, Kubo's formula for the conductance acquires a topological interpretation in terms of Chern numbers and their non-commutative analog, the Fredholm Indices.
J. Func. Analys. 216 (2004), 303-361
ABSTRACT: We study the existence and the continuity properties of the boundary values $(H-\lambda\pm\i0)^{-1}$ of the resolvent of a selfadjoint operator $H$ in the framework of the conjugate operator method initiated by E.\ Mourre. We allow the conjugate operator $A$ to be the generator of a $C_0$-semigroup (finer estimates require $A$ to be maximal symmetric) and we consider situations where the first commutator $[H,\i A]$ is not comparable to $H$. The applications include the spectral theory of zero mass quantum field models.
ABSTRACT: We construct interacting quantum fields in 1+1 space-time dimensions, representing charged or neutral scalar bosons at positive temperature and zero chemical potential. Our work is based on prior work by Klein and Landau and H\o egh-Krohn. Generalized path space methods are used to add a spatially cut-off interaction to the free system, which is described in the Araki-Woods representation. It is shown that the interacting KMS state is normal w.r.t.\ the Araki-Woods representation. The observable algebra and the modular conjugation of the interacting system are shown to be identical to the ones of the free system and the interacting Liouvillean is described in terms of the free Liouvillean and the interaction.
Journal of Statistical Physics {\bf 123}, 585-600 (2006)
ABSTRACT: The ground-state phase diagram of the two-dimensional Falicov-Kimball model with nearest-neighbour and next-nearest-neighbour hoppings has been studied in the perturbative regime where hoppings are small compared with the on-site Coulomb interaction. The phase diagram at fourth-order exhibits a richer structure than the one of the ordinary Falicov-Kimball model.
Math. Nachr. 278 (2005), no. 7-8, 918--931.
ABSTRACT: We consider Schrodinger operators of atoms in a homogeneous magnetic field. We prove that for each fixed value of the pseudomomentum, the corresponding operator has an infinite number of eigenvalues. The asymptotic of the counting function is studied for values of spectral parameter near the bottom of the essential spectrum
Moscow Mathematical Journal, Vol. 3 (1), p.1-17 (2003)
ABSTRACT:
Let $H=-\Delta+V$ be a two-dimensional Schr\"odinger operator
defined on
a domain $\Omega\subset \mathbb R^2$ with Dirichlet boundary conditions.
Suppose that $H$ and $\Omega$ are invariant with respect to translations
in the $x_1$-direction, so that $V(x_1,x_2)=V(x_1+1, x_2)$ and that in addition
$V(x_1, x_2)=V(-x_1, x_2)$ and that $(x_1, x_2)\in\Omega$ implies $(x_1+1,x_2)\in\Omega$
and $(-x_1, x_2)\in\Omega$.
We investigate the associated
Floquet operator $H^{(q)}$, $ 0\le q<1$. In particular we show that the
lowest eigenvalue $\lambda_q$ is simple for $q\neq 1/2$ and strictly
increasing in $q$ for $0< q< 1/2$ and that the associated complex valued
eigenfunction $u_q$ has empty zero set.
For the Dirichlet realization of the Aharonov Bohm Hamiltonian in
an annulus-like
domain with an
axis of symmetry,
$$H_{A,V}=(i \partial_{x_1}+A_1)^2 +(i \partial_{x_2}+A_2)^2 +V\;,$$
we assume that the magnetic
field ${\rm curl}\, A$ vanishes identically and we obtain similar results, where the parameter $q$ is now replaced by
the $\frac{1}{2\pi}$- flux through the hole.
Series on Partial Differential Equations and Applications-Vol.1- World Scientific (2002).
ABSTRACT:
In this book we shall analyze with techniques
coming mainly from partial differential equations (PDE) and of semi-classical analysis
problems coming from statistical mechanics.
Our main object of analysis is a (family of) measure(s) representing the
probability of presence of $m$ particles in interaction and having the form
$$
d\mu^{(m)}:=Z(m,h)^{-1}\exp -\frac {\Phi^{(m)} (X)}{h} \;d X $$
($m\in \mathbb N$) where
Annales de l'ENS t. 37, p.105-170 (2004).
ABSTRACT: In a recent paper Lu and Pan have analyzed the asymptotic behavior, in the semi-clasical regime, of the ground state energy of the Neumann realization of the Schrodinger operator in the case of dimension~$3$. Although these results are rather satisfactory when the magnetic field is non constant and satisfies some generic conditions, they are not sufficient in the case of a constant magnetic field for understanding phenomena like the onset of superconductivity and more accurate results should be obtained. In the two-dimensional case, the effects due to the curvature of the boundary were predicted by a formal analysis of Bernoff-Sternberg and finally proved by the joint efforts of Lu-Pan, Del Pino-Fellmer-Sternberg and Helffer-Morame. Our aim is to analyze similar effects in dimension $3$. As known from physicists and roughly analyzed by Lu-Pan, it turns out that the results depend on the geometry of the boundary especially at the points where the magnetic field is tangent at the boundary. We present here the analog of the Bernoff-Sternberg conjecture (also formulated in a different form by Pan) and prove it in the generic situation.
Springer Lecture Notes in Mathematics 1862 (2005)
ABSTRACT: This review text is an expanded version of informal notes prepared by the first author for a minicourse of eight hours given at Rennes, for the workshop ``{\'e}quations cin{\'e}tiques, hypoellipticit{\'e} et Laplacien de Witten'' organized in February 2003 by the second author. It was substancially completed after the workshop by the two authors with the aim of showing applications to the Fokker-Planck operator in continuation of the work by H{\'e}rau-Nier. We try to present the strong links existing (at the technical level) between the techniques developed originally for the hypoellipticity and the spectral techniques appearing in the analysis of the Schr{\"o}dinger operator or more specifically of the Witten Laplacian. In the context of kinetic equations, it will be shown how the Witten Laplacian, which can itself present degenerate ellipticity, appears as the natural elliptic model for the hypoelliptic H{\"o}rmander type drift diffusion operator involved in the Fokker-Planck equation.
Actes du colloque d'Hammamet (September 2003).
ABSTRACT: Ces derni\`eres ann\'ees, les estimations hypoelliptiques ont connu une nouvelle jeunesse en liaison avec des questions provenant de la th\'eorie cin\'etique des gaz. Dans cette direction de nombreux auteurs ont en effet eu besoin de d\'emontrer des estimations maximales pour en d\'eduire la compacit\'e de l'op\'erateur de Fokker-Planck et avoir des estimations sur la r\'esolvante permettant d'aborder la question du retour \`a l'\'equilibre. Dans un article tr\`es r\'ecent, F.~H\'erau et F.~Nier (inspir\'es par des calculs explicites du livre de Risken) ont mis en \'evidence les liens \'etroits entre ces questions et des questions analogues pour un laplacien de Witten. L'\'etude de ces liens est poursuivie et syst\'ematis\'ee dans un livre en pr\'eparation \'ecrit en collaboration avec F.~Nier dont nous allons pr\'esenter quelques aspects ici en pr\'esentant parfois un \'eclairage diff\'erent sur un probl\`eme qui laisse encore beaucoup de conjectures non r\'esolues. PAPER
J. Phys. A: Math. Gen. 36, 9755-9778 (2003)
ABSTRACT: We investigate the ground state properties of a gas of interacting particles confined in an external potential in three dimensions and subject to rotation around an axis of symmetry. We consider the so-called Gross-Pitaevskii (GP) limit of a dilute gas. Analyzing both the absolute and the bosonic ground state of the system we show, in particular, their different behavior for a certain range of parameters. This parameter range is determined by the question whether the rotational symmetry in the minimizer of the GP functional is broken or not. For the absolute ground state, we prove that in the GP limit a modified GP functional depending on density matrices correctly describes the energy and reduced density matrices, independent of symmetry breaking. For the bosonic ground state this holds true if and only if the symmetry is unbroken.
Reviews in Mathematical Physics 15, 10 (2003), 1219-1253.
ABSTRACT: In this paper we study neutral atoms of nuclear charge Z in a strong constant magnetic field of strength B. We improve the bound on confinement to lowest Landau band given by Lieb, Solovej and Yngvason. This permits to calculate the asymptotic form of the quantum current of atoms in the parameter region $Z^{4/3} \ll B \ll Z^{46/27}$.
Journal of Functional Analysis Vol 208/2 pp 446-481.
ABSTRACT: We study the semi-classical trace formula at a critical energy level for a $h$-pseudo-differential operator whose principal symbol has a unique non-degenerate critical point for that energy. This leads to the study of Hamiltonian systems near equilibrium and near the non-zero periods of the linearized flow. The contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals. The new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface.
Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta) vol.111. No. 1. 49-53 (2004)
ABSTRACT: We use the concept of the exchange hole introduced by Slater to bound the energy of atoms, molecules, and other systems interacting by Coulomb forces from below by one-particle Hamiltonians with an effective screening potential and an exchange hole around each electron. Interestingly enough the optimal size of the exchange hole is smaller than Slater proposed: the best lower bound is obtained when the exchange hole carries charge $1/2$ instead of $1$. To highlight the quality of our estimate we show that the Dirac exchange energy with a slightly different constant bounds the exchange-correlation energy from below, an estimate previously derived by Lieb and Oxford
To appear in Ann. H. Poincare.
ABSTRACT: n recent years,constructive field techniques and the method of renormalization group around extended singularities have been applied to the weak coupling regime of the Anderson Model. It has allowed to clarify the relationship between this model and the theory of random matrices. We review this situation and the current program to analyze in detail the density of states and Green's functions of this model using the supersymmetric formalism.
ABSTRACT: We study the stationary states of a quantum mechanical system describing an atom coupled to black-body radiation at positive temperature. The stationary states of the non-interacting system are given by product states, where the particle is in a bound state corresponding to an eigenvalue of the particle Hamiltonian, and the field is in its equilibrium state. We show that if Fermi's Golden Rule predicts that a stationary state disintegrates after coupling to the radiation field then it is unstable, provided the coupling constant is sufficiently small (depending on the temperature). The result is proven by analyzing the spectrum of the thermal Hamiltonian (Liouvillian) of the system within the framework of $W^*$-dynamical systems. A key element of our spectral analysis is the positive commutator method.
A garden of quanta, 345-363, World Sci. Publishing, River Edge, NJ, 2003
ABSTRACT: We describe some results concerning non-relativistic quantum systems at positive temperature and density confined to macroscopically large (infinite) regions of physical space. In a general discussion of time-dependent thermodynamic processes, we propose a definition of (time-dependent) entropy which has natural physical properties. In this general framework, we show that for a class of time-dependent perturbations approaching a limiting perturbation, the system exhibits return to equilibrium. If the perturbation is periodic then the system approaches a time-periodic state for large times. We apply these results to a reservoir of non-relativistic non-interacting fermions.
ABSTRACT: The general theory of simple transport processes between quantum mechanical reservoirs is reviewed and extended. We focus on thermoelectric phenomena, involving exchange of energy and particles. The theory is illustrated on the example of two reservoirs of free fermions coupled through a local interaction. We construct a stationary state and determine energy and particle currents with the help of a convergent perturbation series. We explicitly calculate several interesting quantities to lowest order, such as the entropy production, the resistance, and the heat conductivity. Convergence of the perturbation series allows us to prove that they are strictly positive under suitable smallness and regularity assumptions on the interaction between the reservoirs.
Mathematical Physics, Analysis and Geometry 7, no. 3, 239-287 (2004)
ABSTRACT: In the context of an idealized model describing an atom coupled to black-body radiation at a sufficiently high positive temperature, we show that the atom will end up being ionized in the limit of large times. Mathematically, this is translated into the statement that the coupled system does not have any time-translation invariant state of positive (asymptotic) temperature, and that the expectation value of an arbitrary finite-dimensional projection in an arbitrary initial state of positive (asymptotic) temperature tends to zero, as time tends to infinity. These results are formulated within the general framework of $W^*$-dynamical systems, and the proofs are based on Mourre's theory of positive commutators and a new virial theorem. Results on the so-called standard form of a von Neumann algebra play an important role in our analysis.
Comm. Math. Phys. 252 (2004), no. 1-3, 415--476.
ABSTRACT: Scattering in a model of a massive quantum-mechanical particle, an ``electron'', interacting with massless, relativistic bosons, ``photons'', is studied. The interaction term in the Hamiltonian of our model describes emission and absorption of ``photons'' by the ``electron''; but ``electron-positron'' pair production is suppressed. An ultraviolet cutoff and an (arbitrarily small, but fixed) infrared cutoff are imposed on the interaction term. In a range of energies where the propagation speed of the dressed ``electron'' is strictly smaller than the speed of light, unitarity of the scattering matrix is proven, provided the coupling constant is small enough; (asymptotic completeness of Compton scattering). The proof combines a construction of dressed one--electron states with propagation estimates for the ``electron'' and the ``photons''.
Commun. Math. Phys. 252, 485 - 534, (2004)
ABSTRACT: We continue the study of the two-component charged Bose gas initiated by Dyson in 1967. He showed that the ground state energy for $N$ particles is at least as negative as $-CN^{7/5}$ for large $N$ and this power law was verified by a lower bound found by Conlon, Lieb and Yau in 1988. Dyson conjectured that the exact constant $C$ was given by a mean-field minimization problem that used, as input, Foldy's calculation (using Bogolubov's 1947 formalism) for the one-component gas. Earlier we showed that Foldy's calculation is exact insofar as a lower bound of his form was obtained. In this paper we do the same thing for Dyson's conjecture. The two-component case is considerably more difficult because the gas is very non-homogeneous in its ground state.
Published in J.Math.Phys 44(2003),1961-1970
ABSTRACT: We consider a Pauli-Fierz Hamiltonian for a particle coupled to a photon field. We discuss the effects of increase of the binding energy and enhanced binding through coupling to a onphoton field, and prove that both effects are the results of the existence of the ground state of the self-energy operator with total momentum $P=0$.
Asymptotic Analysis 41, p. 215-258 (2005)
ABSTRACT: The superconducting properties of a sample submitted to an external magnetic field are mathematically described by the minimizers of the Ginzburg-Landau's functional. The analysis of the Hessian of the functional leads to estimate the fundamental state for the Schr\"odinger operator with intense magnetic field for which the superconductivity appears. So we are interested in the asymptotic behavior of the energy for the Schr\"odinger operator with a magnetic field. A lot of papers have been devoted to this problem, we can quote the works of Bernoff-Sternberg, Lu-Pan, Helffer-Mohamed. These papers deal with estimates of the energy in a regular domain and our goal is to establish similar results in a domain with corners. Although this problem is often mentioned in the physical literature, there are very few mathematical papers. We only know the contributions by Pan and Jadallah which deal with very particular domains like a square or a quarter plane. The Physicists Brosens, Devreese, Fomin, Moshchalkov, Schweigert and Peeters propose a non optimal upper bound for the energy. Here, we present a more rigourous analysis and give an asymptotics of the smallest eigenvalue of the operator in a sector $\Omega_\alpha$ of angle $\alpha$ when $\alpha$ is closed to 0, an estimate for the eigenfunctions and we use these results to study the fundamental state in the semi-classical case. A first version of this work was published by The Royal Swedish Academy of Sciences in 2003; some points are clarified and improved here.
To be published in : Proceedings of ICMP2003 (Plenary talk)
ABSTRACT: Relativistic effects are important in the electrons' dynamics and bound state energies in heavy atoms and molecules. When the nucleii involved are heavily charged, the velocities of the electrons of the inner layers are quite large, and so nonrelativistic modelling will lead to important errors. The usual strategies to address this issue are : either use nonrelativistic models together with relativistic corrections, or use relativistic models based on the Dirac operator.
Journal of Functional Analysis, Vol. 207, pp. 68--110 (2004)
ABSTRACT:
We investigate the self-energy of one electron coupled to a
quantized radiation field by extending the ideas developed in Hainzl
(Ann. IHP, 2003). We fix an arbitrary cut-off parameter $\Lambda$ and
recover the $\alpha^2$-term of the self-energy, where $\alpha$ is the
coupling parameter representing the fine structure constant. Thereby we
develop a method which allows to expand the
self-energy up to {\it any power \/} of $\alpha$. This implies that
perturbation theory in $\alpha$ is correct if $\Lambda$ is fix.
As an immediate consequence we obtain enhanced binding for
electrons.
Journal of Differential Equations, Vol. 205, pp. 253--269 (2004).
ABSTRACT: By variational methods, we prove the inequality \[ \int_\R u''{}^2\,dx-\int_\R u''\,u^2\,dx\geq I\,\int_\R u^4\,dx \] for all $u\in L^4(\R)$ such that $u''\in L^2(\R)$ and for some constant $I\in (-9/64,-1/4)$. This inequality is connected to Lieb-Thirring type problems and has interesting scaling properties. The best constant is achieved by sign changing minimizers of a problem on periodic functions, but does not depend on the period. Moreover, we completely characterize the minimizers of the periodic problem.
ABSTRACT: For a system of $n$ interacting electrons moving in the background of a ``homogeneous'' potential, we show that, if the single particle Hamiltonian admits a density of states, so does the interacting Hamiltonian. Moreover this integrated density of states coincides with that of the free electron Hamiltonian.
ABSTRACT: A non-linear functional $Q[u,v]$ is given that governs the loss, respectively gain, of (doubly degenerate) eigenvalues of fourth order differential operators $L = \partial^4 + \partial\,u\,\partial + v$ on the line. Apart from factorizing $L$ as $A^{*}A + E_{0}$, providing several explicit examples, and deriving various relations between $u$, $v$ and eigenfunctions of $L$, we find $u$ and $v$ such that $L$ is isospectral to the free operator $L_{0} = \partial^{4}$ up to one (multiplicity 2) eigenvalue $E_{0} < 0$. Not unexpectedly, this choice of $u$, $v$ leads to exact solutions of the corresponding time-dependent PDE's.
ABSTRACT: The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrödinger operator is essentially supported by $[0,\infty)$. Our main theorem states that this property is preserved for slowly decaying potentials provided that there are some oscillations with respect to one of the variables.
Journ. Math. Phys. 46, 033512 (2005).
ABSTRACT: We study spectral properties of linear operators. The main tool is the so-called Level Shift Operator, which expresses the effects of 2nd order perturbation theory on the point spectrum. We give estimates for the location of the spectrum in a selected region and approximate formulas for the corresponding spectral projections. Presented results are quite robust---they are applicable even if the spectrum in this region has infinite multiplicity.
New J. Phys. 5, 44.1-44.8 (2003)
ABSTRACT: We study the de Haas-van Alphen oscillations in the magnetization of the Hofstadter model.Near a split band the magnetization is a rapidly oscillating function of the Fermi energy with lip-shaped envelopes.For generic magnetic fields this structure appears on all scales and provides a thermodynamic fingerprint of the fractal properties of the model.The analysis applies equally well to the two dual interpretations of the Hofstadter model and the nature of the duality transformation is elucidated.
J. Funct. Analysis 220 (2), 424-459 (2005)
ABSTRACT: In the Nelson model particles interact through a scalar massless field. For hydrogen-like atoms there is a nucleus of infinite mass and charge $Ze$, $Z > 0$, fixed at the origin and an electron of mass $m$ and charge $e$. This system forms a bound state with binding energy $E_{\rm bin} = me^4Z^2/2$ to leading order in $e$. We investigate the radiative corrections to the binding energy and prove upper and lower bounds which imply that $ E_{\rm bin} = me^4Z^2/2 + c_0 e^6 + \Ow(e^7 \ln e)$ with explicit coefficient $c_0$ and independent of the ultraviolet cutoff. $c_0$ can be computed by perturbation theory, which however is only formal since for the Nelson Hamiltonian the smallest eigenvalue sits exactly at the bottom of the continuous spectrum.
Comm. Pure and App. Math. 57, 528-561 (2004)
ABSTRACT: We introduce a mathematical setup for charge transport in quantum pumps con-nected to a number of external leads.It is proven that under the rather general assumption on the Hamiltonian describing the system,in the adiabatic limit,the current through the pump is given by a formula of Buttiker,Pretre,and Thomas, relating it to the frozen S matrix and its time derivative.
Rev.Math.Phys. 16, 1-28, (2004)
ABSTRACT:
J. Funct. Anal. 203, 93-148 (2003)
ABSTRACT:
J. Stat.Phys. 116, 591-628 (2004)
ABSTRACT:
Comm. Math. Phys. 255, 131-159, (2005)
ABSTRACT: We study the energy of relativistic electrons and positrons interacting via the second quantized Coulomb potential in the field of a nucleus of charge $Z$ within the Hartree-Fock approximation. We show that the associated functional has a minimizer. In addition, all minimizers are purely electronic states, they are projections, and fulfill the no-pair Dirac-Fock equations.
Adv. Theor. and Math. Phys. 7, 145--204 (2003)
ABSTRACT: We study approximate solutions to the Schrödinger equation $i\epsi\partial\psi_t(x)/\partial t = H(x,-i\epsi\nabla_x) \psi_t(x)$ with the Hamiltonian given as the Weyl quantization of the symbol $H(q,p)$ taking values in the space of bounded operators on the Hilbert space $\Hi_{\rm f}$ of fast ``internal'' degrees of freedom. By assumption $H(q,p)$ has an isolated energy band. Using a method of Nenciu and Sordoni \cite{NS} we prove that interband transitions are suppressed to any order in $\epsi$. As a consequence, associated to that energy band there exists a subspace of $L^2(\mathbb{R}^d,\Hi _{\rm f})$ almost invariant under the unitary time evolution. We develop a systematic perturbation scheme for the computation of effective Hamiltonians which govern approximately the intraband time evolution. As examples for the general perturbation scheme we discuss the Dirac and Born-Oppenheimer type Hamiltonians and we reconsider also the time-adiabatic theory.
Commun. Math. Phys. 242, 547-578 (2003)
ABSTRACT: We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, $\phi(\epsi x)$, and vector potential $A(\epsi x)$, with $x \in \R^d$ and $\epsi \ll 1$. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of $L^2(\R^d)$ and an effective Hamiltonian governing the evolution inside this subspace to all orders in $\epsi$. To leading order the effective Hamiltonian is given through the Peierls substitution. We explicitly compute the first order correction. From a semiclassical analysis of this effective quantum Hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics.
Lecture Notes in Mathematics, Springer-Verlag, 2003
Book, pages: 236
ABSTRACT: In nonrelativistic QED the charge of an electron equals its bare value, whereas the self-energy and the mass have to be renormalized. In our contribution we study perturbative mass renormalization, including second order in the fine structure constant $\alpha$, in the case of a single, spinless electron. As well known, if $m$ denotes the bare mass and $\mass$ the mass computed from the theory, to order $\alpha$ one has $$\frac{\mass}{m} =1+\frac{8\alpha}{3\pi} \log(1+\half (\Lambda/m))+O(\alpha^2)$$ which suggests that $\mass/m=(\Lambda/m)^{8\alpha/3\pi}$ for small $\alpha$. If correct, in order $\alpha^2$ the leading term should be $\displaystyle \half ((8\alpha/3\pi)\log(\Lambda/m))^2$. To check this point we expand $\mass/m$ to order $\alpha^2$. The result is $\sqrt{\Lambda/m}$ as leading term, suggesting a more complicated dependence of $m_{\mathrm{eff}}$ on $m$.
J. Funct. Anal. 203, 44--92 (2003)
ABSTRACT:
Comm. Math. Phys. 249 (2004), 29-78
ABSTRACT: We study the spectral theory of massless Pauli-Fierz models using an extension of the Mourre method.We prove the local finiteness of point spectrum and a limiting absorption principle away from the eigenvalues for an arbitrary coupling constant. In addition we show that the expectation value of the number operator is finite on all eigenvectors.
Comm. Math. Phys. 255, (2005) 183-227.
ABSTRACT: We show that electronic wave functions Psi of atoms and molecules have a representation Psi=F*phi, where F is an explicit universal factor, locally Lipschitz, and independent of the eigenvalue and the solution Psi itself, and phi has locally bounded second derivatives. This representation turns out to be optimal as can already be demonstrated with the help of hydrogenic wave functions. The proofs of these results are, in an essential way, based on a new elliptic regularity result which is of independent interest. Some identities that can be interpreted as cusp conditions for second order derivatives of Psi are derived.
J. Funct. Anal. 217 (2004), no. 1, 79--102.
ABSTRACT: We study the semi-classical trace formula at a critical energy level for a $h$-pseudo-differential operator on $\mathbb{R}^{n}$ whose principal symbol has a totally degenerate critical point for that energy. We compute the contribution to the trace formula of isolated non-extremum critical points under a condition of "real principal type". The new contribution to the trace formula is valid for all time in a compact subset of $\mathbb{R}$ but the result is modest since we have restrictions on the dimension.
Mathematical results in quantum mechanics (Taxco, 2001), vol 307, pages 143--148
ABSTRACT: We prove that the electronic density of atomic and molecular eigenfunctions is smooth away from the nuclei. The result is proved without decay assumptions on the eigenfunctions.
submitted for publication in Commun. Math. Phys.
ABSTRACT:We study the behavior of solitary-wave solutions of some generalized nonlinear Schrodinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We construct solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton's equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.
Geom. Funct. Anal., vol. 12 , 989-1017 (2002)
ABSTRACT:
Ann. Inst. H. Poincare Anal. Non Lineaire, vol. 20, 145-181 (2003)
ABSTRACT:
Comm. Math. Phys., 207(3):557-587 (1999)
ABSTRACT:
Rep. Math. Phys. 54 (2004) 169-199
ABSTRACT: We study Pauli-Fierz Hamiltonians---self-adjoint operators describing a small quantum system interacting with a bosonic field. Using quadratic form techniques, we extend the results of Derezi\'nski-G\'erard and G\'erard about the self-adjointness, the location of the essential spectrum and the existence of a ground state to a large class of Pauli-Fierz Hamiltonians.
Comm. Math. Phys. 257 (2005), no. 3, 515-562.
ABSTRACT: Abstract: According to Dirac's ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator $D^0$. In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane, we consider the Bogoliubov-Dirac-Fock model, which is derived from QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is a solution of a self-consistent equation. We show the existence of a minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum.
Journal of Mathematical Physics, Vol. 45(11), pp. 4174--4185 (2004).
ABSTRACT: We consider a spinless particle coupled to a quantized Bose field and show that such a system has a ground state for two classes of short-range potentials which are alone too weak to have a zero-energy resonance.
Published in J. Math. Phys., vol. 46, 042106, 28 pp, (2005)
ABSTRACT: In a recent series of papers, Avron and his collaborators shed new light on the question of quantum transport in mesoscopic samples coupled to particle reservoirs by semi-infinite leads. They rigorously treat the case when the sample undergoes an adiabatic evolution thus generating a current through the leads, and prove the so called BPT formula. Using a discrete model, we complement their work by giving a rigorous proof of the Landauer-Büttiker formula, which deals with the current generated by an adiabatic evolution on the leads. As it is well known in physics, these formulae link the conductance coefficients for such systems to the $S$-matrix of the associated scattering problem. As an application, we discuss the resonant transport through a quantum dot. The single charge tunneling processes are mediated by extended edge states simultaneosly localized near several leads.
Comm. Math. Phys. 247 (2004), no. 2, 513--526.
ABSTRACT: We study the semi-classical trace formula at a critical energy level for an $h$-pseudo-differential operator on $mathbb{R}^n$ whose principal symbol has a totally degenerate critical point for that energy. This problem is studied for a large time behavior and under the hypothesis that the principal symbol of the operator has a local extremum at the critical point.
Ann. Henri Poincare 6 (1) (2005), 85-102.
ABSTRACT: We study the ground state solutions of the Dirac-Fock model in the case of weak electronic repulsion, using bifurcation theory. They are solutions of a min-max problem. Then we investigate a max-min problem coming from the electron-positron field theory of Bach-Barbaroux-Helffer-Siedentop. We show that given a radially symmetric nuclear charge, the ground state of Dirac-Fock solves this max-min problem for certain numbers of electrons. But we also exhibit a situation in which the max-min level does not correspond to a solution of the Dirac-Fock equations together with its associated self-consistent projector.
ABSTRACT: We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on $mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local minimum. The main result, which computes the contribution of this equilibrium, is valid for all time in a compact and establishes the existence of a total asymptotic expansion whose top order coefficient depends only on the germ of the potential at the critical point.
Calc. Var. Partial Differential Equations 24, no. 3, (2005) 341-376.
ABSTRACT: We study the Ginzburg-Landau functional in the parameter regime describing `Type II superconductors'. In the exact regime considered minimizers are localized to the boundary - i.e. the sample is only superconducting in the boundary region. Depending on the relative size of different parameters we describe the concentration behaviour and give leading order energy asymptotics. This generalizes previous results by Lu and Pan, Helffer and Pan, and Pan.
ABSTRACT: Defining a most economical parametrization of time-dependent B-> rho^\pm pi^\mp decays, including a measurable phase alpha_{eff} which equals the weak phase alpha in the limit of vanishing penguin amplitudes, we propose two ways for determining alpha in this processes. We explain the limitation of one method, assuming only that two relevant tree amplitudes factorize and that their relative strong phase, delta_t, is negligible. The other method, based on broken flavor SU(3), permits a determination of alpha in B^0-> rho^\pm pi^\mp in an overconstrained system using also rate measurements of B^{0,+}-> K^* pi and B^{0,+}->rho K. Current data are shown to restrict two ratios of penguin and tree amplitudes, r_\pm, to a narrow range around 0.2, and to imply an upper bound |alpha_{eff} - alpha| < 15 degrees. Assuming that delta_t is much smaller than 90 degrees, we find alpha =(93\pm 17) degrees and (102 \pm 19) degrees using BABAR and BELLE results for B(t)-> rho^\pm pi^mp. Avoiding this assumption for completeness, we demonstrate the reduction of discrete ambiguities in alpha with increased statistics, and show that SU(3) breaking effects are effectively second order in r_\pm.
Phys.Lett.B596:107-115,2004
ABSTRACT: Flavor SU(3) is used to constrain the coefficients of $\sin\Delta mt$ and $\cos\Delta mt$ in the time-dependent CP asymmetry of $B^0 \to \eta' K_S$. Correlated bounds in the $(S_{\eta' K}, C_{\eta' K})$ plane are derived, by using recent rate measurements of $B^0$ decays into $K^+ K^-, \pi^0\pi^0, \pi^0\eta, \pi^0\eta', \eta\eta, \eta\eta', \eta'\eta'$. Stringent bounds are obtained when assuming a single SU(3) singlet amplitude and when neglecting annihilation-type amplitudes.
Phys.Rev.D69:113003,2004
ABSTRACT: It is shown that the weak phase gamma=arg(-V_{ud}V^*_{ub}V_{cb}V_{cd}^*) can be determined using only untagged decays B/Bbar--> D K_S. In order to reduce the uncertainty in gamma, we suggest combining information from B^{+-}--> DK^{+-} and from untagged B^0 decays, where the D meson is observed in common decay modes. Theoretical assumptions, which may further reduce the statistical error, are also discussed.
Phys.Rev.D68:094012,2003
ABSTRACT: We investigate the decay mechanisms in the Ds+ --> omega pi+ and Ds+ --> rho0 pi+ transitions. The naive factorization ansatz predicts vanishing amplitude for the Ds+ --> omega pi+ decay, while the Ds+ --> rho0 pi+ decay amplitude does have an annihilation contribution also in this limit. Both decays can proceed through intermediate states of hidden strangeness, e.g. K, K*, which we estimate in this paper. These contributions can explain the experimental value for the Ds+ --> omega pi+ decay rate, which no longer can be viewed as a clean signature of the annihilation decay of Ds+. The combination of the \pi(1300) pole dominated annihilation contribution and the internal K, K* exchange can saturate present experimental upper bound on Ds+ --> rho0 pi+ decay rate, which is therefore expected to be within the experimental reach. Finally, the proposed mechanism of hidden strangeness FSI constitutes only a small correction to the Cabibbo allowed decay rates Ds--> K K*, phi pi, which are well described already in the factorization approximation.
Phys. Rev. Lett. 98, 186001, (2004)
ABSTRACT: Efficient swimming at low Reynolds numbers is a major concern of microbots. To compare the efficiencies of different swimmers we introduce the notion of ``swimming drag coefficient'' which allows for the ranking of swimmers. We find the optimal swimmer within a certain class of two dimensional swimmers using conformal mappings techniques.
ABSTRACT: Coined quantum walks may be interpreted as the motion in position space of a quantum particle with a spin degree of freedom; the dynamics are determined by iterating a unitary transformation which is the product of a spin transformation and a translation conditional on the spin state. Coined quantum walks on Z d can be treated as special cases of coined quantum walks on R d . We study quantum walks on R d and prove that the sequence of rescaled probability distributions in position space associated to the unitary evolution of the particle converges to a limit distribution.
ABSTRACT: We formulate and prove a general weak limit theorem for quantum random walks in one and more dimensions. With Xn denoting position at time n, we show that Xn=n converges weakly as n ! 1 to a certain distribution which is absolutely continuous and of bounded support. The proof is rigorous and makes use of Fourier transform methods. This approach simpli?es and extends certain preceding derivations valid in one dimension that make use of combinatorial and path integral methods.
To appear in AIHP Probability and Statistics
ABSTRACT: We prove invariance principles for phase separation lines in the two dimensional nearest neighbour Ising model up to the critical temperature and for connectivity lines in the general context of high temperature finite range ferromagnetic Ising models.
Contribution to "Quantum Noise", edited by Yu. V. Nazarov and Ya. M. Blanter (Kluwer)
ABSTRACT: We present a novel derivation of the original Levitov formula, for the statistics of charge transported between electron reservoirs. This is done by proving a trace formula, which relates certain traces in Fock space to single particle determinants. Using the present approach we find in addition several generalizations, such as a corresponding formula for Bosons.
ABSTRACT: We demonstrate, in an elementary manner, that given a partition of the single particle Hilbert space into orthogonal subspaces, a Fermi sea may be factored into pairs of entangled modes, similar to a BCS state. We derive expressions for the entropy and for the particle number fluctuations of a subspace of a Fermi sea, at zero and finite temperatures, and relate these by a lower bound on the entropy. As an application we investigate analytically and numerically these quantities for electrons in the lowest Landau level of a quantum Hall sample.
Journal of Functional Analysis 220: 243--264 (2005)
ABSTRACT: We investigate the low-lying spectrum of Witten-Laplacians on forms of arbitrary degree in the semi-classical limit and uniformly in the space dimension. We show that under suitable assumptions implying that the phase function has a unique minimum one obtains a number of bands of discrete eigenvalues at the bottom of the spectrum. Moreover we are able to count the number of eigenvalues in each band. We apply our results to certain sequences of Schroedinger operators having strictly convex potentials and show that some well-known results of semi-classical analysis hold also uniformly in the dimension
Lett. Math. Phys. 70 (2004), no. 3, 249--257
ABSTRACT: For the Pauli-Fierz operator with a short range potential we study the binding threshold as a function of the fine structure constant $\alpha$ an show that it converges to the binding threshold for the Schrödinger operator in the small $\alpha$ limit.
Documenta Matematica Vol. 9 (2004) p. 283-299.
ABSTRACT: Let $H(\Om_0)=-\Delta+V$ be a Schr\"odinger operator on a bounded domain $\Om_0\subset \mathbb R^d$ ($d\geq 2$) with Dirichlet boundary condition. Suppose that $\Om_\ell$ ($\ell \in \{1,\dots,k\}$) are some pairwise disjoint subsets of $\Om_0$ and that $H(\Om_\ell)$ are the corresponding Schr\"odinger operators again with Dirichlet boundary condition. We investigate the relations between the spectrum of $H(\Om_0)$ and the spectra of the $H(\Om_\ell)$. In particular, we derive some inequalities for the associated spectral counting functions which can be interpreted as generalizations of Courant's nodal theorem. For the case where equality is achieved we prove converse results. In particular, we use potential theoretic methods to relate the $\Om_\ell$ to the nodal domains of some eigenfunction of $H(\Omega_0)$.
Proceedings of the Symposium on Scattering and Spectral Theory. Matematica Contemporanea (Brazilian Mathematical Society), Vol. 26, p. 41-86 (2004).
ABSTRACT: We are interested in the exponentially small eigenvalues of the semiclassical Witten Laplacian on $0$-forms $$ \Delta_{f,h}^{(0)}=-h^{2}\Delta +\left|\nabla f(x)\right|^{2}-h\Delta f(x). $$ We shall consider this operator on $\Omega$ which is either a connected compact Riemannian manifold or $\rz^{n}$. The function $f$ will be a Morse function and when $\Omega$ is a compact manifold for example it is known that there are exactly $m_{0}$ eigenvalues in some interval $[0,e^{-\alpha/h}]$ for $h>0$ small enough, where $m_{0}$ is the number of local minima. Moreover the same result holds for Witten Laplacians on $p$-forms if $m_{p}$ denotes the number of critical points of index $p$.\\ Our purpose is to derive very accurate asymptotic formulas for the $m_{0}$ first eigenvalues of $\Delta_{f,h}^{(0)}$. A similar problem was considered by many authors via a probabilistic approach and more recently by A.~Bovier, V.~Gayrard and M.~Klein, but our result is optimal under rather generic assumptions.
To appear in Memoires de la SMF (2006)
ABSTRACT: This article is a continuation of previous works by Bovier-Eckhoff-Gayrard-Klein, Bovier-Gayrard-Klein and Helffer-Klein-Nier. The main object is the analysis of the small eigenvalues (as $h\ar 0$) of the Laplacian attached to the quadratic form $$ C_0^\infty(\Omega)\ni v \mapsto h^2 \,\int_{\Omega}\left|\nabla v (x)\right|^2\;e^{-2f(x)/h}~dx\;, $$ where $\Omega$ is a bounded connected open set with $C^\infty$-boundary and $f$ is a Morse function on $M=\overline{\Omega}$ . The previous works were devoted to the case of a manifold $M$ which is compact but without boundary or $\rz^n$. Our aim is here to analyze the case with boundary. After the introduction of a Witten cohomology complex adapted to the case with boundary, we give a very accurate asymptotics for the exponentially small eigenvalues. In particular, we analyze the effect of the boundary in the asymptotics.
To appear in Proceedings of QMath9, Giens, France 2004, eds J. Asch and A. Joye
ABSTRACT: We present a construction of the algebra of operators and the Hilbert space for a quantum massless field in 1+1 dimensions.
Eur. Phys. J. B 44, 501-507 (2005)
ABSTRACT: Lieb and Schupp have obtained a number of ground-state properties for frustrated Heisenberg models. The basic tool used was certain version of ``spin-reflection positivity'' method. One group of these results is related to singlet nature of ground state. It needs an assumption of reflection symmetry present in the system. In this paper, it is shown that analogous results hold also for other symmetries (inversion etc.). The second Lieb-Schupp result is matrix inequality, which imply inequalities between ground-state energies of certain systems. In the paper, the Lieb-Schupp inequality is applied to relate ground-state energies of various systems: spin chains, ladders and multidimensional lattices.
Phys. Rev. A, 70, 023612-(1-12) (2004)
ABSTRACT: Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice. We present a model of this phenomenon which we are able to analyze rigorously. The system is a hard core lattice gas at half-filling and the optical lattice is modeled by a periodic potential of strength $\lambda$. For small $\lambda$ and temperature, BEC is proved to occur, while at large $\lambda$ or temperature there is no BEC. At large $\lambda$ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase. The interparticle interaction is essential for this transition, which occurs even in the ground state. Surprisingly, the condensation is always into the $p=0$ mode in this model, although the density itself has the periodicity of the imposed potential.
Commun. Math. Phys. 266, 797-818 2006
ABSTRACT: We prove upper bounds on the ground state energies of the one- and two-component charged Bose gases. The upper bound for the one-component gas agrees with the high density asymptotic formula proposed by L. Foldy in 1961. The upper bound for the two-component gas agrees in the large particle number limit with the asymptotic formula conjectured by F. Dyson in 1967. Matching asymptotic lower bounds for these systems were proved in references \cite{LS} and \cite{LS2}. The formulas of Foldy and Dyson which are based on Bogolubov's pairing theory have thus been validated.
Ann. Henri Poincare, 6, 247-267, 2005
ABSTRACT: We consider the supersymmetric quantum mechanical system which is obtained by dimensionally reducing d=6, N=1 supersymmetric gauge theory with gauge group U(1) and a single charged hypermultiplet. Using the deformation method and ideas introduced by Porrati and Rozenberg, we present a detailed proof of the existence of a normalizable ground state for this system.
Few-Body Systems vol 34, 155-161 (2004)
ABSTRACT: Excitons in carbon nanotubes may be modeled by two oppositely charged particles living on the surface of a cylinder. We derive three one-dimensional effective Hamiltonians which become exact as the radius of the cylinder vanishes. Two of them are solvable.
Phys. Rev. A 71, 012503 (2005)
ABSTRACT: We present the derivation of the effective higher-order Hamiltonian, which gives $m \alpha^6$ contribution to energy levels of an arbitrary light atom. The derivation is based on the Foldy-Wouthuysen transformation of the one-particle Dirac Hamiltonian followed by perturbative expansion of the many particle Green function. The obtained results can be used for the high precision calculation of relativistic effects in atomic systems.
Commun. Math. Phys. 261, 549 - 568
ABSTRACT: We study systems containing electrons and nuclei. Based on the fact that the Thermodynamic limit exists for systems with Dirichlet boundary conditions, we prove that the same limit is obtained if one imposes other boundary conditions such as Neumann, periodic, or elastic boundary conditions. The result is proven for all limiting sequences of domains which are obtained by scaling a bounded open set, with smooth boundary, except for isolated edges and corners.
Annales de l'Institut Fourier 56 , no. 1, (2006) 1-67.
ABSTRACT: Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension 2, many papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schroedinger operator with magnetic field have appeared recently. In the present paper we settle one important part of this question by proving an asymptotic expansion to all orders for low lying eigenvalues for generic domains. The word generic means in this context that the curvature of the boundary of the domain has a unique non-degenerate maximum.
Journ. Math. Phys. 46, 063511 (2005).
ABSTRACT: The weak coupling (van Hove) limit of one parameter groups of contractions is studied by the stationary approach. We show that the resolvent of the properly renormalized and rescaled generator of a contractive semigroup has a limit as the coupling constant goes to zero. This limit is the resolvent of the generator of a certain contractive semigroup. Our results can be viewed as a stationary counterpart to the well known results about the weak coupling limit obtained by the time-dependent approach, due to E. B. Davies. We compare both approaches.
Lett. Math. Phys. 72, 27-38, (2005).
ABSTRACT: Within the framework of local quantum physics we construct a scattering theory of stable, massive particles without assuming mass gaps. This extension of the Haag-Ruelle theory is based on advances in the harmonic analysis of local operators. Our construction is restricted to theories complying with a regularity property introduced by Herbst. The paper concludes with a brief discussion of the status of this assumption.
ABSTRACT: We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disordered-broadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, namely a boundary point between the localization and delocalization regimes, which we prove to converge to the corresponding Landau level as either the magnetic field or the disorder goes to zero.
ABSTRACT: We develop a Hamiltonian theory for a time dispersive and dissipative inhomogeneous medium, as described by a linear response equation respecting causality and power dissipation. The canonical Hamiltonian constructed here exactly reproduces the original dissipative evolution after integrating out auxiliary fields. In particular, for a dielectric medium we obtain a simple formula for the Hamiltonian and closed form expressions for the energy density and energy flux involving the auxiliary fields. The developed approach also allows to treat a long standing problem of scattering from a lossy non-spherical obstacle.
ABSTRACT: We consider the edge and bulk conductances for 2D quantum Hall systems in which the Fermi energy falls in a band where bulk states are localized. We show that the resulting quantities are equal, when appropriately defined. An appropriate definition of the edge conductance may be obtained through a suitable time averaging procedure or by including a contribution from states in the localized band. In a further result on the Harper Hamiltonian, we show that this contribution is essential. In an appendix we establish quantized plateaus for the conductance of systems which need not be translation ergodic.
Illinois Jour. Math. vol 49, pp. 893-904 (2005).
ABSTRACT: We prove that the integrated density of states (IDS) associated to a random Schroedinger operator is locally uniformly Hoelder continuous as a function of the disorder parameter lambda. In particular, we obtain convergence of the IDS, as lambda tends to 0, to the IDS for the unperturbed operator at all energies for which the IDS for the unperturbed operator is continuous in energy.
ABSTRACT: We justify the linear response theory for an ergodic Schroedinger operator with magnetic field within the non-interacting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators where the Liouville equation can be solved uniquely. If the Fermi level falls into a region of localization, we recover the well-known Kubo-Streda formula for the quantum Hall conductivity at zero temperature.
J. Stat. Phys. 118 (2005), 199
ABSTRACT: We study linear time dispersive and dissipative systems. Very often such systems are not conservative and the standard spectral theory can not be applied. We develop a mathematically consistent framework allowing (i) to constructively determine if a given time dispersive system can be extended to a conservative one; (ii) to construct that very conservative system -- which we show is essentially unique. We illustrate the method by applying it to the spectral analysis of time dispersive dielectrics and the damped oscillator with retarded friction. In particular, we obtain a conservative extension of the Maxwell equations which is equivalent to the original Maxwell equations for a dispersive and lossy dielectric medium.
ABSTRACT: Hölder continuity, $|N_\lambda(E)-N_\lambda(E')|\le C |E-E'|^\alpha$, with a constant $C$ independent of the disorder strength $\lambda$ is proved for the integrated density of states $N_\lambda(E)$ associated to a discrete random operator $H = H_o + \lambda V$ consisting of a translation invariant hopping matrix $H_o$ and i.i.d. single site potentials $V$ with an absolutely continuous distribution, under a regularity assumption for the hopping term.
ABSTRACT: Algebra and representation theory in modular tensor categories can be combined with tools from topological field theory to obtain a deeper understanding of rational conformal field theories in two dimensions: It allows us to establish the existence of sets of consistent correlation functions, to demonstrate some of their properties in a model-independent manner, and to derive explicit expressions for OPE coefficients and coefficients of partition functions in terms of invariants of links in three-manifolds. We show that a Morita class of (symmetric special) Frobenius algebras $A$ in a modular tensor category $\calc$ encodes all data needed to describe the correlators. A Morita-invariant formulation is provided by module categories over $\calc$. Together with a bimodule-valued fiber functor, the system (tensor category + module category) can be described by a weak Hopf algebra. The Picard group of the category $\calc$ can be used to construct examples of symmetric special Frobenius algebras. The Picard group of the category of $A$-bimodules describes the internal symmetries of the theory and allows one to identify generalized Kramers-Wannier dualities.
Seminaire EDP, Ecole Polytechnique, 2003-2004
ABSTRACT: We discuss the Hartree equation arising in the mean-field limit of large systems of bosons and explain its importance within the class of nonlinear Schroedinger equations. Of special interest to us is the Hartree equation with focusing nonlinearity (attractive two-body interactions). Rigorous results for the Hartree equation are presented concerning: 1) its derivation from the quantum theory of large systems of bosons, 2) existence and stability of Hartree solitons, and 3) its point-particle (Newtonian) limit. Some open problems are described.
ABSTRACT: We study tight-binding models of itinerant electrons in two different bands, with effective on-site interactions expressing Coulomb repulsion and Hund's rule. We prove that, for sufficiently large on-site exchange anisotropy, all ground states show metallic ferromagnetism: They exhibit a macroscopic magnetization, a macroscopic fraction of the electrons is spatially delocalized, and there is no energy gap for kinetic excitations.
Phys.Lett. B596 (2004) 156-162
ABSTRACT: In contrast to the usual representations of of the Poincar\'e group of finite spin or helicity the Wigner representations of mass zero and infinite spin are known to be incompatible with pointlike localized quantum fields. We present here a construction of quantum fields associated with these representations that are localized in semi-infinite, space-like strings. The construction is based on concepts outside the realm of Lagrangian quantization with the potential for further applications.
To appear in Reports on Mathematical Physics (2005)
ABSTRACT: One of von Neumann's motivations for developing the theory of operator algebras and his and Murray's 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type ${\rm III}_{1}$ factor in Connes' classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e., the way they are embedded into each other.
Rep. Math. Phys. {\bf 55}, 135--147, (2005)
Phys. Rev. Lett. Vol. 94, 80401 (2005)
ABSTRACT: The validity of substituting a c-number $z$ for the $k=0$ mode operator $a_0$ is established rigorously in full generality, thereby verifying one aspect of Bogoliubov's 1947 theory. This substitution not only yields the correct value of thermodynamic quantities like the pressure or ground state energy, but also the value of $|z|^2$ that maximizes the partition function equals the true amount of condensation in the presence of a gauge-symmetry breaking term -- a point that had previously been elusive.
p. 199-215 in Mathematical Physics of Quantum Mechanics Lecture notes in physics, eds. J. Asch and Alain Joye, Springer 2006. Proceedings of QMath9, Giens, France, Sept. 12--16, 2004
ABSTRACT: One of the most remarkable recent developments in the study of ultracold Bose gases is the observation of a reversible transition from a Bose Einstein condensate to a state composed of localized atoms as the strength of a periodic, optical trapping potential is varied. In \cite{ALSSY} a model of this phenomenon has been analyzed rigorously. The gas is a hard core lattice gas and the optical lattice is modeled by a periodic potential of strength $\lambda$. For small $\lambda$ and temperature Bose-Einstein condensation (BEC) is proved to occur, while at large $\lambda$ BEC disappears, even in the ground state, which is a Mott-insulator state with a characteristic gap. The inter-particle interaction is essential for this effect. This contribution gives a pedagogical survey of these results.
Physical Review Letters, vol. 88 #170409 (2002)
ABSTRACT: The ground state of bosonic atoms in a trap has been shown experimentally to display Bose-Einstein condensation (BEC). We prove this fact theoretically for bosons with two-body repulsive interaction potentials in the dilute limit, starting from the basic Schroedinger equation; the condensation is 100% into the state that minimizes the Gross-Pitaevskii energy functional. This is the first rigorous proof of BEC in a physically realistic, continuum model.
ABSTRACT: We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on $mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result, which establishes the contribution of the associated equilibrium in the trace formula, is valid for all time in a compact subset of $mathbb{R}$ and includes the singularity in $t=0$. For these new contributions the asymptotic expansion involves the logarithm of the parameter $h$. Depending on an explicit arithmetic condition on the dimension and the order of the critical point, this logarithmic contribution can appear in the leading term.
New J. of Physics 7, 234, (2005)
ABSTRACT: The swimming of a pair of spherical bladders that can change their shape is elementary at small Reynolds numbers. If large strokes are allowed then the swimmer has superior efficiency to other models of artificial swimmers. The change of shape resembles the wriggling motion known as {\it metaboly} of certain protozoa.
Ann. Henri Poincare 4 , 1083-1099 (2003).
ABSTRACT: The Jansen-Hess operator is an approximate (pseudo-)relativistic no-pair Hamiltonian in the Furry picture which is used in the physics literature to describe heavy atoms. Within the single-particle Coulomb model we prove that their energy, and thus the resulting self-adjoint operator and its spectrum, is positive for $ Z \leq 114 $.
Math. Phys. Electron. J. 8, Paper 3, 30 pp. (electronic) (2002).
ABSTRACT: It is shown that the essential spectrum of the pseudo-relativistic Dirac operator according to Jansen and Hess which includes the Coulomb potential up to second order, extends from $mc^2$ to infinity when the nuclear charge is below the critical value $Ze^2 \approx 1.006.$ There is also no singular continuous spectrum in that case, and for small $Z$ no embedded eigenvalues. This work is an extension of investigations by Evans, Perry and Siedentop on the Brown-Ravenhall operator which is of first order in the potential. It is based on the fact, recently proven by Brummelhuis, Siedentop and Stockmeyer, that the spectrum of the Jansen-Hess operator is bounded from below for subcritical charges $Z$.
ABSTRACT: By means of an exponential unitary transformation scheme borrowed from the study of quantum lattice systems, the Dirac operator of a one-electron ion is transformed into a pseudo-relativistic operator which easily allows for the elimination of the positron degrees of freedom. This operator is block-diagonal with respect to the projection onto the positive (respective negative) spectral subspace of the free Dirac operator, to a fixed order in the strength of the electron-nucleus Coulomb potential. It is demonstrated that this transformation scheme is unitarily equivalent to the one introduced by Douglas and Kroll, and that the pseudo-relativistic operators of (up to) first and second order in the potential strength agree with the Brown-Ravenhall and the Jansen-Hess operator, respectively. The transformation scheme is successively applied to two-electron and N-electron systems in a Coulomb central field. Moreover, the investigations of Evans, Perry and Siedentop and of Balinsky and Evans, concerning the spectral properties of the single-particle Brown-Ravenhall operator for subcritical potential strength, are extended to the Jansen-Hess operator. Both the single-particle and the multiparticle Jansen-Hess operator are investigated.
Physical Review A 71 (2005).
ABSTRACT:By means of a unitary transformation scheme, the Coulomb-Dirac operator for two electrons in a central potential is transformed into a pseudorelativistic operator which allows for the decoupling of the electron and positron degrees of freedom to arbitrary order /n/ in the potential strength. In case of /n/=2, relative boundedness properties and positivity of the resulting operator are shown for subcritical potential strength.
J. Math. Phys., 46, 052307 (2005)
ABSTRACT: In this paper we study the large-Z behaviour of the ground state energy of atoms with electrons having relativistic kinetic energy sqrt(p^2c^2+m^2c^4)-mc^2. We prove that to leading order in Z the energy is the same as in the non-relativistic case, given by (non-relativistic) Thomas-Fermi theory. For the problem to make sense, we keep the product Z*alpha fixed (here alpha is Sommerfeld's fine structure constant), and smaller than, or equal to, 2/pi, which means that as Z tends to infinity, alpha tends to zero.
J. Phys. A: Math. and Gen. 38 (2005), 4483-4499.
ABSTRACT: We study the Bogoliubov-Dirac-Fock model introduced by Chaix and Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989) which is a mean-field theory deduced from no-photon QED. The associated functional is bounded from below. In the presence of an external field, a minimizer, if it exists, is interpreted as the polarized vacuum and it solves a self-consistent equation. In a recent paper math-ph/0403005, we proved the convergence of the iterative fixed-point scheme naturally associated with this equation to a global minimizer of the BDF functional, under some restrictive conditions on the external potential, the ultraviolet cut-off $\Lambda$ and the bare fine structure constant $\alpha$. In the present work, we improve this result by showing the existence of the minimizer by a variational method, for any cut-off $\Lambda$ and without any constraint on the external field. We also study the behaviour of the minimizer as $\Lambda$ goes to infinity and show that the theory is "nullified" in that limit, as predicted first by Landau: the vacuum totally kills the external potential. Therefore the limit case of an infinite cut-off makes no sense both from a physical and mathematical point of view. Finally, we perform a charge and density renormalization scheme applying simultaneously to all orders of the fine structure constant $\alpha$, on a simplified model where the exchange term is neglected.
Lett. Math. Phys. 72 (2005), 99-113.
ABSTRACT: We consider a generalized Dirac-Fock type evolution equation deduced from no-photon Quantum Electrodynamics, which describes the self-consistent time-evolution of relativistic electrons, the observable ones as well as those filling up the Dirac sea. This equation has been originally introduced by Dirac in 1934 in a simplified form. Since we work in a Hartree-Fock type approximation, the elements describing the physical state of the electrons are infinite rank projectors. Using the Bogoliubov-Dirac-Fock formalism, introduced by Chaix-Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989), and recently established by Hainzl-Lewin-Sere, we prove the existence of global-in-time solutions of the considered evolution equation.
Comm. Pure Appl. Math., in press.
ABSTRACT: We study the mean-field approximation of Quantum Electrodynamics, by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normal-ordering or choice of bare electron/positron subspaces. Neglecting photons, we define properly this Hamiltonian in a finite box $[-L/2;L/2)^3$, with periodic boundary conditions and an ultraviolet cut-off $\Lambda$. We then study the limit of the ground state (i.e. the vacuum) energy and of the minimizers as $L$ goes to infinity, in the Hartree-Fock approximation. In case with no external field, we prove that the energy per volume converges and obtain in the limit a translation-invariant projector describing the free Hartree-Fock vacuum. We also define the energy per unit volume of translation-invariant states and prove that the free vacuum is the unique minimizer of this energy. In the presence of an external field, we prove that the difference between the minimum energy and the energy of the free vacuum converges as $L$ goes to infinity. We obtain in the limit the so-called Bogoliubov-Dirac-Fock functional. The Hartree-Fock (polarized) vacuum is a Hilbert-Schmidt perturbation of the free vacuum and it minimizes the Bogoliubov-Dirac-Fock energy.
in Lecture Notes in Mathematics 1882, Open Quantum Systems III, eds S. Attal, A. Joye, C.-A. Pillet, Springer 2006
ABSTRACT: We describe and compare two applications of the Fermi Golden Rule to the study of a small quantum system interacting with a reservoir. The first application is the weak coupling limit for the dynamics reduced to a subsystem. As a result one obtains a Markov completely positive semigroup generated by the so-called Davies generator. The second application is the computation of the Level Shift Operator for the Liouvillean, which is used in the problem of the return to equilibrium. We show that in the thermal case the Davies generator and the Level Shift Operator for the Liouvillean are related by a similarity transformation.
ABSTRACT: We establish spectral estimates at a critical energy level for $h$-pseudors . Via a trace formula, we compute the contribution of isolated (non-extremum) critical points under a condition of "real principal type". The main result holds for all dimensions, for a singularity of any finite order and can be invariantly expressed in term of the geometry of the singularity. When the singularities are not integrable on the energy surface the results are significative since the order w.r.t. $h$ of the spectral distributions are bigger than in the regular setting.
Annales of Henri Poincar$eacute;, 2006, volume 7, number 1, pages 45-58.
ABSTRACT:We show that the Douglas-Kroll block-diagonalization method for the Dirac operator with Coulomb potential is convergent in norm resolvent sense for coupling constant $\gamma$ less than $\gamma_c=0.37758$ which corresponds to atomic number $51$. Moreover, we give an explicit expression for the corresponding block-diagonalized Dirac operator.
New York Journal of Mathematics 11, 225-245 (2005).
ABSTRACT: We shall consider a locally compact groupoid endowed with a Haar system $\nu $ and having proper orbit space. We shall associate to each appropriate cross section $\sigma :G^{\left( 0\right) }\rightarrow G^{F}$ for $d_{F}:G^{F}\rightarrow G^{\left( 0\right) }$ (where $F$ is a Borel subset $G^{\left( 0\right) }$ meeting each orbit exactly once) a $C^{\ast }$-algebra $M_{\sigma }^{\ast }\left( G,\nu \right) $. We shall prove that the C^{\ast }$-algebras $M_{\sigma }^{\ast }\left( G,\nu _{i}\right) $ $i=1,2$ associated with different Haar systems are $\ast $-isomorphic
Comm. Math. Phys. 266, 153-196 (2006)
ABSTRACT: Using recent results by the authors on the spectral asymptotics of the Neumann Laplacian with magnetic field, we give precise estimates on the critical field, $H_{C_3}$, describing the appearance of superconductivity in superconductors of type II. Furthermore, we prove that the local and global definitions of this field coincide. Near $H_{C_3}$ only a small part, near the boundary points where the curvature is maximal, of the sample carries superconductivity. We give precise estimates on the size of this zone and decay estimates in both the normal (to the boundary) and parallel variables.
Rev. Math. Phys, Vol. 17, No. 6 (2005) 669-743
ABSTRACT: We study the long time evolution of a quantum particle interacting with a random potential in the Boltzmann-Grad low density limit. We prove that the phase space density of the quantum evolution defined through the Husimi function converges weakly to a linear Boltzmann equation. The Boltzmann collision kernel is given by the full quantum scattering cross section of the obstacle potential.
Submitted to the Proceedings of the QMath9 Conference. (2005)
ABSTRACT: We consider random Schrödinger equations on $\bR^d$ or $\bZ^d$ for $d\ge 3$ with uncorrelated, identically distributed random potential. Denote by $\lambda$ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$. Suppose that the space and time variables scale as $x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}$ with $0< \kappa \leq \kappa_0$, where $\kappa_0$ is a sufficiently small universal constant. We proved that, in the limit $\lambda \to 0$, the expectation value of the Wigner distribution of $\psi_t$, $\bE W_{\psi_{t}} (x, v)$, converges weakly to a solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum $v$.
ABSTRACT:We consider random Schrodinger equations on ${\bf Z^d}$ for $d\ge 3$ with identically distributed random potential. Denote by $\lambda$ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$. The space and time variables scale as $x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}$ with $0< \kappa < 1/6000$. We prove that, in the limit $\lambda \to 0$, the expectation of the Wigner distribution of $\psi_t$ converges weakly to a solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum $v$. This work is an extension to the lattice case of our previous result in the continuum [ESYI], [ESYII]. Due to the non-convexity of the level surfaces of the dispersion relation, the estimates of several Feynman graphs are more involved.
Accepted for publication in Inventiones Mathematicae
ABSTRACT: We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schrodinger equation in a suitable scaling limit. The result is extended to $k$-particle density matrices for all positive integer $k$.
Doc. Math. 10, 331-356 (2005).
ABSTRACT: A sequence of unitary transformations is applied to the one-electron Dirac operator in an external Coulomb potential such that the resulting operator is of the form $\Lambda_+ A\,\Lambda_+\,+\Lambda_-A\,\Lambda_-\;$ to any given order in the potential strength, where $\Lambda_+$ and $\Lambda_-$ project onto the positive and negative spectral subspaces of the free Dirac operator. To first order, $\Lambda_+ A \,\Lambda_+$ coincides with the Brown-Ravenhall operator. Moreover, there exists a simple relation to the Dirac operator transformed with the help of the Foldy-Wouthuysen technique. By defining the transformation operators as integral operators in Fourier space it is shown that they are well-defined and that the resulting transformed operator is $p$-form bounded. In the case of a modified Coulomb potential, $ V=-\gamma x^{-1+\epsilon},\;\;\epsilon >0,\;$ one can even prove subordinacy of the $n$-th order term in $\gamma$ with respect to the $n-1$st order term for all $n>1$, as well as their $p$-form boundedness with form bound less than one.
ABSTRACT: The localization of the essential spectrum of a relativistic two-electron ion is provided. The analysis is performed with the help of the pseudo-relativistic Brown-Ravenhall operator which is the restriction of the Coulomb-Dirac operator to the electrons' positive spectral subspace.
Physics Letters A, Volume 341, Issues 5-6, 27 June 2005, Pages 473-478
ABSTRACT: We block-diagonalize the Dirac operator D_\gamma approximately to explain that the methods of Douglas, Kroll, and Hess yield convergent schemes to calculate the spectral line of D_\gamma.
ABSTRACT: We study the transversal XY spin-spin correlations for non-equilibrium steady state and prove their spatial exponential decay close to equilibrium.
ABSTRACT: For a quantum particle interacting with a short-range potential, we estimate from below the shift of its binding threshold, which is due to the particle interaction with a quantized radiation field.
Accepted for publication in Comm. Pure Appl. Math. (2005)
ABSTRACT: Consider a system of $N$ bosons on the three dimensional unit torus interacting via a pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, \ldots, x_N)$ denotes the positions of the particles. Suppose that the initial data $\psi_{N,0}$ satisfies the condition \[ \langle \psi_{N,0}, H_N^2 \psi_{N,0} \rangle \leq C N^2 \] where $H_N$ is the Hamiltonian of the Bose system. This condition is satisfied if $\psi_{N,0}= W_N \phi_{N,0}$ where $W_N$ is an approximate ground state to $H_N$ and $\phi_{N,0}$ is regular. Let $\psi_{N,t}$ denote the solution to the Schr\"odinger equation with Hamiltonian $H_N$. Gross and Pitaevskii proposed to model the dynamics of such system by a nonlinear Schr\"odinger equation, the Gross-Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if $u_t$ solves the GP equation, then the family of $k$-particle density matrices $\{ \otimes_k u_t, k\ge 1 \}$ solves the GP hierarchy. We prove that as $N\to \infty$ the limit points of the $k$-particle density matrices of $\psi_{N,t}$ are solutions of the GP hierarchy. The uniqueness of the solutions to this hierarchy remains an open question. Our analysis requires that the $N$ boson dynamics is described by a modified Hamiltonian which cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical inter-particle distance. Our proof can be extended to a modified Hamiltonian which only forbids at least $n$ particles from coming close together, for any fixed $n$.
Lecture Notes in Physics 690 (Springer), p. 403-415 (2006)
ABSTRACT: The aim of this lecture is to present the recent results obtained in collaboration with M.~Klein and F.~Nier on the low lying eigenvalues of the Laplacian attached to the Dirichlet form~: $$ C_0^\infty(\Omega)\ni v \mapsto h^2 \,\int_{\Omega}\left|\nabla v (x)\right|^2\;e^{-2f(x)/h}~dx\;, $$ where $f$ is a $C^\infty$ Morse function on $\overline{\Omega}$ and $h>0$. We give in particular an optimal asymptotics as $h\ar 0$ of the lowest strictly positive eigenvalue, which will hold under generic assumptions. We discuss also some aspects of the proof.
Math. Phys. Anal. Geom., vol 8 pp. 315 - 360 (2006)
ABSTRACT: This paper is devoted to the asymptotics of the density of surfacic states near the spectral edges for a discrete surfacic Anderson model. Two types of spectral edges have to be considered : fluctuating edges and stable edges. Each type has its own type of asymptotics. In the case of fluctuating edges, one obtains Lifshitz tails the parameters of which are given by the initial operator suitably ``reduced'' to the surface. For stable edges, the surface density of states behaves like the surface density of states of a constant (equal to the expectation of the random potential) surface potential. Among the tools used to establish this are the asymptotics of the surface density of states for constant surface potentials.
Mémoires de la SMF, vol 104, {\tt vi}+105 pages (2006)
ABSTRACT: In this paper, we study spectral properties of the one dimensional periodic Schrodinger operator with an adiabatic quasi-periodic perturbation. We show that in certain energy regions the perturbation leads to resonance effects related to the ones observed in the problem of two resonating quantum wells. These effects affect both the geometry and the nature of the spectrum. In particular, they can lead to the intertwining of sequences of intervals containing absolutely continuous spectrum and intervals containing singular spectrum. Moreover, in regions where all of the spectrum is expected to be singular, these effects typically give rise to exponentially small "islands" of absolutely continuous spectrum.
Ann. Scient. ENS. vol 38, pp 889-950 (2005)
ABSTRACT: In this paper, we consider one dimensional adiabatic quasi-periodic Schrodinger operators in the regime of strong resonant tunneling. We show the emergence of a level repulsion phenomenon which is seen to be very naturally related to the local spectral type of the operator: the more singular the spectrum, the weaker the repulsion.
Comm. Math. Phys. vol 267, pp 669-701 (2006)
ABSTRACT: We consider the 2D Landau Hamiltonian $H$ perturbed by a random alloy-type potential, and investigate the Lifshitz tails, i.e. the asymptotic behavior of the corresponding integrated density of states (IDS) near the edges in the spectrum of $H$. If a given edge coincides with a Landau level, we obtain different asymptotic formulae for power-like, exponential sub-Gaussian, and super-Gaussian decay of the one-site potential. If the edge is away from the Landau levels, we impose a rational-flux assumption on the magnetic field, consider compactly supported one-site potentials, and formulate a theorem which is analogous to a result obtained in the case of a vanishing magnetic field.
To appear in Moscow Mathematical Journal
ABSTRACT: We consider two-dimensional Schrodinger operators in bounded domains. Abstractions of nodal sets are introduced and spectral conditions for them ensuring that they are actually zero sets of eigenfunctions are given.
ABSTRACT: We study the Schrodinger operator $(h\mathbf{D}-\mathbf{A})^2$ with periodic magnetic field $B= \text{curl}\,\mathbf{A}$ in an antidot lattice $\Omega_\infty=\R^2\setminus\bigcup_{\alpha\in\Gamma}(U+\alpha)$. Neumann boundary conditions lead to spectrum below $h\inf B$. Under suitable assumptions on a "one-well problem" we prove that this spectrum is localized inside an exponentially small interval in the semi-classical limit $h\rightarrow 0$. For this purpose we construct a basis of the corresponding spectral subspace with natural localization and symmetry properties.
ABSTRACT: We present a mathematically rigorous analysis of the ground state of a dilute, interacting Bose gas in a three-dimensional trap that is strongly confining in one direction so that the system becomes effectively two-dimensional. The parameters involved are the particle number, $N\gg 1$, the two-dimensional extension, $\bar L$, of the gas cloud in the trap, the thickness, $h\ll \bar L$ of the trap, and the scattering length $a$ of the interaction potential. Our analysis starts from the full many-body Hamiltonian with an interaction potential that is assumed to be repulsive, radially symmetric and of short range, but otherwise arbitrary. In particular, hard cores are allowed. Under the premisses that the confining energy, $\sim 1/h^2$, is much larger than the internal energy per particle, and $a/h\to 0$, we prove that the system can be treated as a gas of two-dimensional bosons with scattering length $a_{\rm 2D}= h\exp(-(\hbox{\rm const.)}h/a)$. In the parameter region where $a/h\ll |\ln(\bar\rho h^2)|^{-1}$, with $\bar\rho\sim N/\bar L^2$ the mean density, the system is described by a two-dimensional Gross-Pitaevskii density functional with coupling parameter $\sim Na/h$. If $|\ln(\bar\rho h^2)|^{-1}\lesssim a/h$ the coupling parameter is $\sim N |\ln(\bar\rho h^2)|^{-1}$ and thus independent of $a$. In both cases Bose-Einstein condensation in the ground state holds, provided the coupling parameter stays bounded.
to appear in Comm. Mah. Phys,
In Mathematical Physics Studies Vol. 2, Perspectives in Analysis Essays in Honor of Lennart Carleson's 75th Birthday, Michael Benedicks, Peter W. Jones, Stanislav Smirnov (Eds.), Springer Verlag Berlin Heidelberg 2005 ISBN 3-540-30432-0
ABSTRACT: This article gives a detailed presentation of the authors' recent results on the ground state properties of the Bose gas. It is a much expanded version of a talk given by one of the authors (E.H.L.) at the conference "Perspectives in Analysis" at the KTH, Stockholm, June 23, 2003. It is based on, but supersedes, the article math-ph/0204027.
Series: Oberwolfach Seminars, Vol. 34, 2005, VIII, 208 p., ISBN: 3-7643-7336-9. A Birkhauser textbook.
ABSTRACT: This book is an extended version of the material presented in a course for graduate students and PostDocs in Oberwolfach, June 2004.
Phys. stat. sol. C {\bf 3}, p. 199-203 (2006)
ABSTRACT: Finite-temperature properties of the Falicov-Kimball model on the square lattice have been studied in the perturbative regime, i.e. in the case: $t\slash{}U\ll 1$, where $t$ is the hopping constant and $U$ denotes the Coulomb interaction strength. In our study, we have determined the phase diagram of the model in the second-order of the perturbation theory, where the antiferromagnetic Ising model in the magnetic field emerges, and partially in the fourth-order, where it constitutes the Ising model with more complicated frustrated antiferromagnetic interactions. The Monte Carlo method was employed, which proved its accuracy in recent analyses of other spin models, like Ashkin-Teller model. To determine the type of ordering and phase boundaries, the behavior of Binder cumulants, based on the order parameters under consideration, was analyzed.
ABSTRACT: We consider the Brown--Ravenhall model of a relativistic atom with N electrons and a nucleus of charge Z and prove the existence of an infinite number of discrete eigenvalues for N <= Z. As an intermediate result we prove a HVZ-type theorem for these systems.
ABSTRACT: We propose three effective Hamiltonians which approximate atoms in very strong homogeneous magnetic fields $B$ modelled by the Pauli Hamiltonian, with fixed total angular momentum with respect to magnetic field axis. All three Hamiltonians describe $N$ electrons and a fixed nucleus where the Coulomb interaction has been replaced by $B$-dependent one-dimensional effective (vector valued) potentials but without magnetic field. Two of them are solvable in at least the one electron case. We briefly sketch how these Hamiltonians can be used to analyse the bottom of the spectrum of such atoms.
ABSTRACT: We consider the hydrogen molecular ion $H^+_2$ in the presence of a strong homogeneous magnetic field. We determine the leading asymptotic behavior for the equilibrium distance between the nuclei of this molecule in the limit when the strength of the magnetic field goes to infinity.
ABSTRACT: On the bosonic Fock space, a family of Bogoliubov transformations corresponding to a strongly continuous one-parameter group of symplectic maps $R(t)$ is considered. Under suitable assumptions on the generator $A$ of this group, which guarantee that the induced representations of CCR are unitarily equivalent for all time $t$, it is known that the unitary operator $U_{nat}(t)$ which implement this transformation gives a projective unitary representation of $R(t)$. Under rather general assumptions on the generator $A$, we prove that the corresponding Bogoliubov transformations can be implemented by a one-parameter group $U(t)$ of unitary operators. The generator of $U(t)$ will be called a Bogoliubov Hamiltonian. We will introduce two kinds of Bogoliubov Hamiltonians (type I and II) and give conditions so that they are well defined.
to appear in Journ. Math. Phys.
Reference: J. Phys. A Math. Gen. 39 85-98 (2006)
ABSTRACT: Barbaroux, Esteban and Séré have investigated the relation between Mittleman's max-min principle and the solutions of the Dirac-Fock equations. Their comparison is valid in the non-relativistic limit, but it does not contain quantitative estimates. We generalize their result of non-agreement in the case of one electron and show that this non-agreement holds for the physical value of the fine structure constant if /Z/ < 42 (molybdenum).
Archive for Rational Mechanics and Analysis Publisher: Springer Berlin / Heidelberg ISSN: 0003-9527 (Paper) 1432-0673 (Online) DOI: 10.1007/s00205-006-0016-6
ABSTRACT: We consider atoms with closed shells, i.e., the electron number $N$ is $2,\ 8,\ 10,...$, and weak electron-electron interaction. Then there exists a unique solution $\gamma$ of the Dirac-Fock equations $[D_{g,\alpha}^{(\gamma)},\gamma]=0$ with the additional property that $\gamma$ is the orthogonal projector onto the first $N$ positive eigenvalues of the Dirac-Fock operator $D_{g,\alpha}^{(\gamma)}$. Moreover, $\gamma$ minimizes the energy of the relativistic electron-positron field in Hartree-Fock approximation, if the splitting of $\gH:=L^2(\rz^3)\otimes \cz^4$ into electron and positron subspace, is chosen self-consistently, i.e., the projection onto the electron-subspace is given by the positive spectral projection of $D_{g,\alpha}^{(\gamma)}$. For fixed electron-nucleus coupling constant $g:=\alpha Z$ we give quantitative estimates on the maximal value of the fine structure constant $\alpha$ for which the existence can be guaranteed.
Markov Proc. Related Fields, vol 11, 177--188 (2005)
ABSTRACT: We prove the thermodynamic limit for generalized susceptibilities of a quantum gas in the grandcanonical ensembles.
Published in J. Math. Phys. vol 47, nr. 1, 23 pp, (2006)
ABSTRACT: This paper is the first in a series revisiting the Faraday effect, or more generally, the theory of electronic quantum transport/optical response in bulk media in the presence of a constant magnetic field. The independent electron approximation is assumed. At zero temperature and zero frequency, if the Fermi energy lies in a spectral gap, we rigorously prove the Widom-Streda formula. For free electrons, the transverse conductivity can be explicitly computed and coincides with the classical result. In the general case, using magnetic perturbation
Annales Henri Poincare, in press.
ABSTRACT: In this paper, we continue the mathematical study of adiabatic chemical reactions, started in a previous work (Ann. Henri Poincare 5, 477-521, 2004). We consider a molecule with one free atom, the latter having two distinct possible stable positions. We then look for a mountain pass point between these two local minima in the non-relativistic Schroedinger framework. We prove the existence of a mountain pass point without any assumption on the molecules at infinity, improving our previous results for this model. This critical point is interpreted as a transition state in Quantum Chemistry.
ABSTRACT: We review Evans' contributions to the spectral theory of operators describing relativistic particle systems. We will concentrate on no-pair operators and recent extensions of that work
ABSTRACT: We consider second quantized homogeneous Bose gas in a large cubic box with periodic boundary conditions,at zero temperature, and in the grand canonical setting (the chemical potential $\mu$ is fixed, the number of particles can vary). We investigate upper bounds on the infimum of the energy for a fixed total momentum $\kk$ given by the expectation value at one-particle excitations over a squeezed state. We show that the results of the Bogoliubov approach (usually derived heuristically) coincide with the results of the first iteration of our method (which leads to rigorous upper bounds).
in Lecture Notes in Physics 695 ``Large Coulomb Systems'', ed. J. Derezinski andH. Siedentop, Springer 2006
ABSTRACT: Lecture notes of a minicourse given at the Summer School on Large Coulomb Systems - QED in Nordfjordeid, 2003, devoted to representations of the CCR and CAR. Quasifree states, the Araki-Woods and Araki-Wyss representations, and the lattice of von Neumann algebras in a bosonic/fermionic Fock space are discussed in detail.
to appear in the proceedings of the conference Operator Algebras and Mathematical Physics 3 in Bucharest, 2005
ABSTRACT: For a class of negative slowly decaying potentials including the attractive Coulombic one we study the classical scattering theory in the low-energy regime. We construct a (continuous) family of classical orbits parametrized by initial position $x\in \R^d$, final direction $\omega\in S^{d-1}$ of escape (to infinity) and the energy $\lambda\geq 0$, yielding a complete classification of the set of outgoing scattering orbits. The construction is given in the outgoing part of phase-space (a similar construction may be done in the incoming part of phase-space). For fixed $\omega\in S^{d-1}$ and $\lambda\geq 0$ the collection of constructed orbits constitutes a smooth manifold that we show is Lagrangian. The family of those Lagrangians can be used to study the quantum mechanical scattering theory in the low-energy regime for the class of potentials considered here. We devote this study to a subsequent paper.
To appear in Journ. Math. Phys.
ABSTRACT: We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a ``small subsystem'' and an infinite dimensional one -- a ``reservoir''. The operator, which we call a ``Friedrichs Hamiltonian'', has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before \cite{AFL} in more complicated models and where they were called "stochastic limit".
J. Phys. A: Math. Gen. *39* No 33 (18 August 2006) 10405-10414
ABSTRACT: It is shown that the ground state energy of heavy atoms is, to leading order, given by the non-relativistic Thomas-Fermi energy. The proof is based on the relativistic Hamiltonian of Brown and Ravenhall which is derived from quantum electrodynamics yielding energy levels correctly up to order $\alpha^2$Ry.
ABSTRACT: In the Dirac operator framework we characterize and estimate the ground state energy of relativistic hydrogenic atoms in a constant magnetic field and describe the asymptotic regime corresponding to a large field strength using relativistic Landau levels. We also define and estimate a critical magnetic field beyond which stability is lost.
Accepted for publication in Ann. Henri Poincaré.
ABSTRACT: We investigate regularity properties of molecular one-electron densities rho near the nuclei. In particular we derive a representation rho(x)=mu(x)*(e^F(x)) with an explicit function F, only depending on the nuclear charges and the positions of the nuclei, such that mu belongs to C^{1,1}(R^3), i.e., mu has locally essentially bounded second derivatives. An example constructed using Hydrogenic eigenfunctions shows that this regularity result is sharp. For atomic eigenfunctions which are either even or odd with respect to inversion in the origin, we prove that mu is even C^{2,\alpha}(R^3) for all alpha in (0,1). Placing one nucleus at the origin we study rho in polar coordinates x=r*omega and investigate rho'(r,omega) and rho''(r,omega) for fixed omega as r tends to zero. We prove non-isotropic cusp conditions of first and second order, which generalize Kato's classical result.
accepted for publication in Adv. in Math.
ABSTRACT: We study confinement of the ground state of atoms in strong magnetic fields to different subspaces related to the lowest Landau band. The results obtained allow us to calculate the quantum current in the entire semiclassical region $B \ll Z^3$.
ABSTRACT: We study electron densities of eigenfunctions of atomic Schroedinger operators. We prove the existence of rho~'''(0), the third derivative of the spherically averaged atomic density rho~ at the nucleus. For eigenfunctions with corresponding eigenvalue below the essential spectrum we obtain the bound rho~'''(0) \leq -(7/12)Z^3 rho~(0), where Z denotes the nuclear charge. This bound is optimal.
ABSTRACT: We investigate properties of the spherically averaged atomic one-electron density rho~(r). For a rho~ which stems from a physical ground state we prove that rho~ > 0. We also give exponentially decreasing lower bounds to rho~ in the case when the eigenvalue is below the corresponding essential spectrum.
ABSTRACT: We consider the Neumann Laplacian with constant magnetic field on a regular domain. Let $B$ be the strength of the magnetic field, and let $\lambda_1(B)$ be the first eigenvalue of the magnetic Neumann Laplacian on the domain. It is proved that $B \mapsto \lambda_1(B)$ is monotone increasing for large $B$. Combined with the results of \cite{FournaisHelffer3}, this implies that all the `third' critical fields for strongly Type II superconductors coincide.
Accepted to the Proceedings of the 60-th birthday conference of Barry Simon (2006).
ABSTRACT: We present a review on the recent developments concerning rigorous mathematical results on Schroedinger operators with magnetic fields. This paper is dedicated to the sixtieth birthday of Barry Simon.
ABSTRACT: We consider random Schroedinger equations on ${\bf R^d}$ for $d\ge 3$ with a homogeneous Anderson-Poisson type random potential. Denote by $\lambda$ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$. The space and time variables scale as $x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}$ with $0< \kappa < \kappa_0(d)$. We prove that, in the limit $\lambda \to 0$, the expectation of the Wigner distribution of $\psi_t$ converges weakly to the solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data. The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the non-recollision graphs and prove that the amplitude of the {\it non-ladder} diagrams is smaller than their ``naive size" by an extra $\lambda^c$ factor per non-(anti)ladder vertex for some $c > 0$. This is the first rigorous result showing that the improvement over the naive estimates on the Feynman graphs grows as a power of the small parameter with the exponent depending linearly on the number of vertices. This estimate allows us to prove the convergence of the perturbation series.
Accepted to Commun. Math. Phys. (2006)
ABSTRACT: We consider random Schr\"odinger equations on ${\bf R}^d$ for $d\ge 3$ with a homogeneous Anderson-Poisson type random potential. Denote by $\lambda$ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$. The space and time variables scale as $x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}$ with $0< \kappa < \kappa_0(d)$. We prove that, in the limit $\lambda \to 0$, the expectation of the Wigner distribution of $\psi_t$ converges weakly to the solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data. The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper [ESYI] the analysis of the non-repetition diagrams was presented. In this paper we complete the proof by estimating the recollision diagrams and showing that the main terms, i.e. the ladder diagrams with renormalized propagator, converge to the heat equation.
ABSTRACT: We prove $L^p$-bounds on the Fourier transform of measures $\mu$ supported on two dimensional surfaces. Our method allows to consider surfaces whose Gauss curvature vanishes on a one-dimensional submanifold. Under a certain non-degeneracy condition, we prove that $\widehat \mu\in L^{4+\beta}$, $\beta>0$, and we give a logarithmically divergent bound on the $L^4$-norm. We use this latter bound to estimate almost singular integrals involving the dispersion relation, $e(p)= \sum_1^3 [1-\cos p_j]$, of the discrete Laplace operator on the cubic lattice. We briefly explain our motivation for this bound originating in the theory of random Schr\"odinger operators.
ABSTRACT: Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $x=(x_1, \ldots, x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $\psi_{N,t}$ be the solution to the Schr\"odinger equation. Suppose that the initial data $\psi_{N,0}$ satisfies the energy condition $$ \langle \psi_{N,0}, H_N^k \psi_{N,0} \rangle \leq C^k N^k \; $$ for $k=1,2,\ldots $. We also assume that the $k$-particle density matrices of the initial state are asymptotically factorized as $N\to\infty$. We prove that the $k$-particle density matrices of $\psi_{N,t}$ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schr\"odinger equation with the coupling constant given by the scattering length of the potential $V$. We also prove the same conclusion if the energy condition holds only for $k=1$ but the factorization of $\psi_{N,0}$ is assumed in a stronger sense.
Arch. Ration. Mech. Anal. 179 (2006), no. 2, 265-283
ABSTRACT: We consider the dynamics of $N$ boson systems interacting through a pair potential $N^{-1} V_a(x_i-x_j)$ where $V_a (x) = a^{-3} V (x/a)$. We denote the solution to the $N$-particle Schrodinger Equation by $\psi_{N, t}$. Recall that the Gross-Pitaevskii (GP) Equation is a nonlinear Schr\"odinger Equation and the GP Hierarchy is an infinite BBGKY Hierarchy of equations so that if $u_t$ solves the GP Equation, then the family of $k$-particle density matrices $\{ \otimes_k u_t, k\ge 1 \}$ solves the GP Hierarchy. Under the assumption that $a = N^{-\epsilon}$ for $0 < \epsilon < 3/5$, we prove that as $N\to \infty$ the limit points of the $k$-particle density matrices of $\psi_{N,t}$ are solutions of the GP Hierarchy with the coupling constant in the nonlinear term of the GP Equation given by $\int V(x) dx$. The uniqueness of the solutions to this hierarchy remains an open question.
Phys. Rev. Lett., *96, *130602 (2006)
ABSTRACT: We propose a quantum theory of swimming for swimmers that are small relative to the coherence length of the medium. The quantum swimming equation is derived from known results on quantum pumps. For a one-dimensional Fermi gas at zero temperature we find that swimming is topological: The distance covered in one swimming stroke is quantized in half integer multiples of the Fermi wave length. Moreover, one can swim without dissipation.
Phys.Rev. Lett., *96, *130501 (2006)
ABSTRACT: Tomographic analysis demonstrates that the polarization state of pairs of photons emitted from a biexciton decay cascade becomes entangled when spectral filtering is applied. The measured density matrix of the photon pair satisfies the Peres criterion for entanglement by more than 3 standard deviations of the experimental uncertainty and violates Bell's inequality. We show that the spectral filtering erases the ``which path'' information contained in the photons color and that the remanent information in the quantum dot degrees of freedom is negligible.
Phys.Rev. Lett., *96, *130501 (2006)
ABSTRACT: When basic tools of quantum information are applied to the quantum tomography data presented in Nature 439, 179 (2006), none of their devices appears to be a source of entangled photons.
New J. Phys. *8* (2006) 68, math-ph/0602053
ABSTRACT: We study the swimming of non-relativistic deformable bodies in (empty) static curved spaces. We focus on the case where the ambient geometry allows for rigid body motions. In this case the swimming equations turn out to be geometric. For a small swimmer, the swimming distance in one stroke is determined by the Riemann curvature times certain moments of the swimmer.
Phys. Rev. lett. in press
ABSTRACT: We consider the Casimir interaction between (non absorbing) dielectric bodies or conductors. Our main result is a proof that the Casimir force between two bodies related by reflection is always attractive, independent of the exact form of the bodies or dielectric properties. Apart from being a fundamental property of fields, the theorem and it's corollaries also rule out a class of suggestions to obtain repulsive forces, the most well known of these being the two hemisphere repulsion suggestion and its relatives.
ABSTRACT: Recently the block-diagonalization of Dirac-operators was investigated from a mathematical point of view in the one-particle case. We extend this result to the $N$-particle case. This leads to a perturbative realization of the Furry picture in the $N$-particle two-spinor space.
ABSTRACT: We prove the convergence in certain weighted spaces in momentum space of eigenfunctions of H = T-lambda*V as the energy goes to an energy threshold. We do this for three choices of kinetic energy T, namely the non-relativistic Schr"odinger operator, the pseudorelativistc operator sqrt{-\Delta+m^2}-m, and the Dirac operator.
Few-Body Systems vol 38, nr. 2-4, 125--131 (2006)
ABSTRACT: We consider three one dimensional quantum, charged and spinless particles interacting through delta potentials. We derive sufficient conditions which guarantee the existence of at least one bound state.
J. Math. Phys. vol 47, 083511 (2006) (25 pp)
ABSTRACT: In this work we study the diamagnetic properties of a perfect quantum gas in the presence of a constant magnetic field of intensity $B$. We investigate the Gibbs semigroup associated to the one particle operator at finite volume, and study its Taylor series with respect to the field parameter $\omega:= eB/c$ in different topologies. This allows us to prove the existence of the thermodynamic limit for the pressure and for all its derivatives with respect to $\omega$ (the so-called generalized susceptibilities).
To appear in Ann. Henri Poincaré.
ABSTRACT: We analyse the low lying spectrum of a model of excitons in carbon nanotubes. Consider two particles with an attractive Coulomb self-interaction, placed on an infinitely long cylinder. If the cylinder radius becomes small, the low lying spectrum is well described by a one-dimensional effective Hamiltonian which is exactly solvable.
J. Math. Phys. 47 (2006) 033506.
ABSTRACT: The condition for $E = 0$ to be an eigenvalue of the operator $\sqrt{-\Delta + m^2} -m + \lambda V$ is obtained through the use of the Birman-Schwinger principle (Theorem 3.2). By setting $E = -\alpha^2$ and using the analyticity of the corresponding Birman-Schwinger kernel (Theorem 3.1), the series development of $\lambda^{-1}$ is obtained up to second order on $\alpha$ (Theorem 4.1).
Phys. Rev. A 71, 053605, 2005
ABSTRACT: Recent developments in the physics of low density trapped gases make it worthwhile to verify old, well known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground state energy of a dilute gas of $N$ fermions interacting with a short-range, positive potential of scattering length $a$. For spin 1/2 fermions, this is $E \sim E^0 + (\hbar^2/2m) 2 \pi N \rho a$, where $E^0$ is the energy of the non-interacting system and $\rho$ is the density. A similar formula holds in 2D, with $\rho a$ replaced by $\rho /|\ln(\rho a^2)|$. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.
to appear in Complex Analysis and Operator Theory
Abstract: Mourre's commutator theory is a powerful tool to study the continuous spectrum of self-adjoint operators and to develop scattering theory. We propose a new approach of its main result, namely the derivation of the limiting absorption principle from a so called Mourre estimate. We provide a new interpretation of this result.
Proceedings "23-emes Journees Equations aux derivees partielles"
Abstract: La theorie de Mourre est un outil puissant pour etudier le spectre continu d'operateurs auto-adjoints et pour developper une theorie de la diffusion. Dans cet expose nous proposons d'un nouveau regard sur la theorie de Mourre en donnant une nouvelle approche du resultat principal de la theorie~: le principe d'aborption limite (PAL) obtenu a partir de l'estimation de Mourre. Nous donnons alors une nouvelle interpretation de ce resultat. Cet expose a aussi pour but d'etre une introduction a la theorie.
Abstract : The Bogoliubov-Dirac-Fock (BDF) model is the mean-field approximation of no-photon Quantum Electrodynamics. The present paper is devoted to the study of the minimization of the BDF energy functional \emph{under a charge constraint}. An associated minimizer, if it exists, will usually represent the ground state of a system of $N$ electrons interacting with the Dirac sea, in an external electrostatic field generated by one or several fixed nuclei. We prove that such a minimizer exists when a binding (HVZ-type) condition holds. We also derive, study and interpret the equation satisfied by such a minimizer. Finally, we provide two regimes in which the binding condition is fulfilled, obtaining the existence of a minimizer in these cases. The first is the weak coupling regime for which the coupling constant $\alpha$ is small whereas $\alpha Z$ and the particle number $N$ are fixed. The second is the non-relativistic regime in which the speed of light tends to infinity (or equivalently $\alpha$ tends to zero) and $Z$, $N$ are fixed. We also prove that the electronic solution converges in the non-relativistic limit towards a Hartree-Fock ground state.
Abstract: We obtain asymptotic expressions for the Green kernels of certain non-translation invariant transition matrices using methods of semiclassical and microlocal analysis. Combined with a result by Bach and M{\o}ller this yields asymptotic formulas for the truncated two-point correlation functions of certain non-translation invariant lattice models of real-valued spins.
General results on the eigenvalues of operators with gaps, arising from both ends of the gaps. Application to Dirac operators.
ABSTRACT: This paper is concerned with an extension and reinterpretation of previous results on the variational characterization of eigenvalues in gaps of the essential spectrum of self-adjoint operators. We state two general abstract results on the existence of eigenvalues in the gap and a continuation principle. Then, these results are applied to Dirac operators in order to characterize simultaneously eigenvalues corresponding to electronic and positronic bound states.
A short review on computational issues arising in relativistic atomic and molecular physics
ABSTRACT: This paper is a short review of computational issues arising when trying to compute electronic wave functions in atoms or molecules containing heavy nuclei. In this case, relativistic effects can play an important role and {\sl ad hoc} models have to be used. The basic operator of this theories is the Dirac operator, which unlike th Schr\"odinger operator, is unbounded both from below. This feature creates dificulties both at the theoretical and the computational levels. In this review we discuss some of these issues.
J. Phys. A: Math. Gen. 39, 7501-7516 (2006)
The pseudorelativistic no-pair Jansen-Hess operator is derived for the case where in addition to the Coulomb potential, an external magnetic field $B$ is permitted. With some restrictions on the vector potential it is shown that this operator is positive provided the strength $\gamma$ of the Coulomb potential is below a critical value ($\gamma_c\leq 0.35$, depending on the magnetic field energy $E_f$). Moreover, for $\gamma <0.32$ and for $B$ tending asymptotically to zero in a weak sense, the essential spectrum is given by $[m,\infty)+E_f.$
Doc. Math. 10, 417-445 (2005)
The HVZ theorem is proven for the pseudorelativistic $N$-electron Jansen-Hess operator $(2\leq N \leq Z)$ which acts on the spinor Hilbert space $\cA(H_1({\Bbb R}^3) \otimes {\Bbb C}^4)^N$ where $\cA$ denotes antisymmetrization with respect to particle exchange. This 'no pair' operator results from the decoupling of the electron and positron degrees of freedom up to second order in the central potential strength $\gamma=Ze^2$.
Based on the HVZ theorem and dilation analyticity of the pseudorelativistic no-pair Jansen-Hess operator, it is shown that for subcritical potential strength ($Z\leq 90)$ the singular continuous spectrum is absent. The bound is slightly higher ($Z\leq 102$) for the Brown-Ravenhall operator whose eigenvalues $\lambda$ are, by the virial theorem, confined to $\lambda <2m$ if $Z\leq 50.$
Journal of Functional Analysis 239 (2006) 310-344
Abstract: A quantum system $\s$ interacts in a successive way with elements $\ee$ of a chain of identical independent quantum subsystems. Each interaction lasts for a duration $\tau$ and is governed by a fixed coupling between $\s$ and $\ee$. We show that the system, initially in any state close to a reference state, approaches a {\it repeated interaction asymptotic state} in the limit of large times. This state is $\tau$--periodic in time and does not depend on the initial state. If the reference state is chosen so that $\s$ and $\ee$ are individually in equilibrium at positive temperatures, then the repeated interaction asymptotic state satisfies an average second law of thermodynamics.
Abstract: The purpose of this paper is extend the notion of morphism of groupoids introduced by Zakrzewski to locally compact $\sigma$-compact groupoids endowed with Haar systems and to use the extension to construct a covariant functor from this category to a category of $C^{\ast}$-algebras in the sense of Woronowicz.
J. Functional Analysis, 238(1):193-220, 2006
Abstract. This paper is devoted to inequalities of Lieb-Thirring type. Let $V$ be a nonnegative potential such that the corresponding Schr\"odinger operator has an unbounded sequence of eigenvalues $(\lambda_i(V))_{i\in{\bf N}^*}$. We prove that there exists a positive constant ${{\cal C}(\gamma)}$, such that, if~$\gamma>d/2$, then $$ \sum_{i\in{\bf N}^*}\left[\lambda_i(V)\right]^{-\gamma}\leq {{\cal C} (\gamma)}\int_{{\bf R}^d}{V^{{d\over 2}-\gamma}}\;dx \eqno(*)$$ and determine the optimal value of ${{\cal C}(\gamma)}$. Such an inequality is interesting for studying the stability of mixed states with occupation numbers. We show how the infimum of $\lambda_1(V)^\gamma\kern-1.5pt\cdot \kern-1.5pt\int_{{\bf R}^d}{V^{{d\over 2}-\gamma}}\,dx$ on all possible potentials $V$, which is a lower bound for $[{{\cal C}(\gamma)}]^{-1}$, corresponds to the optimal constant of a subfamily of Gagliardo-Nirenberg inequalities. This explains how $(*)$ is related to the usual Lieb-Thirring inequality and why all Lieb-Thirring type inequalities can be seen as generalizations of the Gagliardo-Nirenberg inequalities for systems of functions with occupation numbers taken into account. We also state a more general inequality of Lieb-Thirring type $$ \sum_{i\in{\bf N}^*}F(\lambda_i(V))={\rm Trace}\left[ F\left(-\Delta +V\right)\right]\leq\int_{{\bf R}^d}G(V(x))\,dx \eqno(**)$$ where $F$ and $G$ are appropriately related. As a special case corresponding to $F(s)=e^{-s}$, $(**)$ is equivalent to an optimal euclidean logarithmic Sobolev inequality $$ \int_{{\bf R}^d}{\rho\,\log\rho}\;dx+{d\over 2}\,\log(4\pi)\int_{{\bf R}^d}{\rho}\;dx\leq \sum_{i\in{\bf N}^*}{\bf N}u_i\,\log{\bf N}u_i+ \sum_{i\in{\bf N}^*}{\bf N}u_i\int_{{\bf R}^d}{|{\bf N}abla\psi_i|^2}\;dx $$ where $\rho=\sum_{i\in{\bf N}^*}{\bf N}u_i\,|\psi_i|^2$, $({\bf N}u_i) _{i\in{\bf N}}$ is any nonnegative sequence of occupation numbers and $(\psi_i)_{i\in{\bf N}}$ is any sequence of orthonormal $L^2({\bf R}^d)$ functions.
Preprint Ceremade no. 0639, 2006
ABSTRACT: Interpolation inequalities of Gagliardo-Nirenberg type and compactness results for self-adjoint trace-class operators with finite kinetic energy are established. Applying these results to the minimization of various free energy functionals, we determine for instance stationary states of the Hartree problem with temperature corresponding to various statistics.
ABSTRACT: We determine the structure of the spectrum and obtain non-propagation estimates for a class of Toeplitz operators acting on a subset of the lattice $\Z^N$. This class contains the Hamiltonian of the one-dimensional Heisenberg model.
ABSTRACT: We put into evidence graphs with adjacency operator whose singular subspace is prescribed by the kernel of an auxiliary operator. In particular, for a family of graphs called admissible, the singular continuous spectrum is absent and there is at most an eigenvalue located at the origin. Among other examples, the one-dimensional XY model of solid-state physics is covered. The proofs rely on commutators methods.
ABSTRACT: We advocate for the systematic use of a symmetrized definition of time delay in scattering theory. In two-body scattering processes, we show that the symmetrized time delay exists for arbitrary dilated spatial regions symmetric with respect to the origin. It is equal to the usual time delay plus a new contribution, which vanishes in the case of spherical spatial regions. We also prove that the symmetrized time delay is invariant under an appropriate mapping of time reversal. These results are also discussed in the context of classical scattering theory.
Duke Math. J. 110, 161-193 (2001)
ABSTRACT: We exhibit an intermittency phenomenon in quantum dynamics. More precisely we derive new lower bounds for the moments of order {\it p} associated to the state $\psi(t)={\rm e}^{-itH}\psi$ and averaged in time between $0$ and {\it T}. These lower bounds are expressed in terms of generalized fractal dimensions $D^\pm_{\mu_\psi}(1/(1+p/d))$ of the measure $\mu_\psi$ (where {\it d} is the space dimension). This improves previous results, obtained in terms of Hausdorff and Packing dimension.
J. Math. Pure et Appl. 80, 977-1012 (2001)
ABSTRACT: Given a positive probability Borel measure $\mu$, we establish some basic properties of the associated functions $\tau_\mu^\pm(q)$ and of the generalized fractal dimensions $D_\mu^\pm(q)$ for $q\in\R$. We first give the connections between the generalized fractal dimensions, the R\'enyi dimensions and the mean-$q$ dimensions when $q>0$. We then use these relations to prove some regularity properties for $\tau_\mu^\pm(q)$ and $D_\mu^\pm(q)$; we also give some estimates for these functions as well as for their product of convolution. We finally present some calculations for specific examples.
ABSTRACT: When considering a non-relativistic atom coupled to a quantized radiation field, it is natural to require that the model predicts the existence of a ground state. In a recent paper, Griesemer, Lieb and Loss found a condition under which they could prove that the Pauli Fierz model with Coulomb interactions admits a ground state. Here, we consider the Nelson model with Coulombian interactions under the same condition and we show that it does not admit a ground state in the Fock representation, but it does in another, not unitarily equivalent coherent representation.
ABSTRACT: We consider a generalized Ginzburg-Landau model which describes the superconducting properties of a superconducting sample surrounded by a normal material and submitted to an external magnetic field. This model is devoted to the study of proximity effects, usually observed in such situations. We study the stability of normal states in the presence of a large magnetic field and for large values of the Ginzburg-Landau parameter. The obtained results agree with observed effects mentioned in the physical literatur. In certain parameter regimes, our results coincide with those obtained for the `standard' Ginzburg-Landau model, where we mention in particular the two-term expansion for the upper critical field obtained by Helffer-Pan.
J.~Func.~Anal. (in print)
ABSTRACT: This work addresses the problem of infrared mass renormalization for a scalar electron in a translation-invariant model of non-relativistic QED. We assume that the interaction of the electron with the quantized electromagnetic field comprises a fixed ultraviolet regularization and an infrared regularization parametrized by $\sigma>0$. For the value $p=0$ of the conserved total momentum of electron and photon field, bounds on the renormalized mass are established which are uniform in $\sigma\rightarrow0$, and the existence of a ground state is proved. For $|p|>0$ sufficiently small, bounds on the renormalized mass are derived for any fixed $\sigma>0$. A key ingredient of our proofs is the operator-theoretic renormalization group using the isospectral smooth Feshbach map. It provides an explicit, finite algorithm that determines the renormalized electron mass at $p=0$ to any given precision.
Commun.Math.Phys. (2006)
ABSTRACT: In this paper, the groundstate of a nonrelativistic atom, minimally coupled to the quantized radiation field, and its groundstate energy are constructed by an iteration scheme inspired by \cite{Pizzo2003}. This scheme successively removes an infrared cutoff in momentum space and yields a convergent algorithm enabling us to calculate the groundstate and the groundstate energy, to arbitrary order in the feinstructure constant $\alpha \sim 1/137$. In forthcoming papers, we will use our result to re-expand the ground state and, eventually, scattering amplitudes in terms of bare quantities.
ABSTRACT: In this paper, we construct an explicit and constructive algorithm enabling us to calculate the groundstate and the groundstate energy of a nonrelativistic atom minimally coupled to the quantized radiation field up to an error of arbitrary finite order in the feinstructure constant. Because of infrared divergences, which invalidate a straightforward Taylor expansion, an iterative construction is employed to remove the infrared cutoff in photon momentum space and to produce a convergent algorithm.
ABSTRACT:In this paper, we rigorously justify Bohr's frequency condition in atomic spectroscopy. Moreover, we construct an algorithm enabling us to calculate the transition amplitudes for Rayleigh scattering of light at an atom, up to a remainder term of arbitrarily high order in the finestructure constant. Our algorithm is constructive and circumvents the infrared divergences that invalidate standard perturbative theory.
J.Diff.Eqs. (accepted for publication)
ABSTRACT: For a nonrelativistic hydrogen atom minimally coupled to the quantized radiation field we construct the ground state projection $P_\gs$ by a continuous renormalization group scheme as an alternative to the discrete renormalization group scheme recently used by Fr\"ohlich, Pizzo, and the first author \cite{BachFroehlichPizzo2006a}. That is, we construct $P_\gs = \lim_{t \to \infty} P_t$ as the limit of a continuously differentiable family $(P_t)_{t \geq 0}$ of ground state projections of infrared regularized Hamiltonians $H_t$. Using the ODE solved by this family of projections, we show that the norm $\| \dot{P}_t \|$ of their derivative is integrable in $t$ which in turn yields the convergence of $P_t$ by the fundamental theorem of calculus.
Rev.Math.Phys, 18, 519--543 (2006)
ABSTRACT: As a contribution to the study of Hartree-Fock theory we prove rigorously that the Hartree-Fock approximation to the ground state of the $d$-dimensional Hubbard model leads to saturated ferromagnetism when the particle density (more precisely, the chemical potential $\mu$) is small and the coupling constant $U$ is large, but finite. This ferromagnetism contradicts the known fact that there is no magnetization at low density, for any $U$, and thus shows that HF theory is wrong in this case. As in the usual Hartree-Fock theory we restrict attention to Slater determinants that are eigenvectors of the z-component of the total spin, $\mathbb{S}_z = \sum_x n_{x,\uparrow} - n_{x,\downarrow}$, and we find that the choice $2\mathbb{S}_z=N=$ particle number gives the lowest energy at fixed $0 < \mu < 4d$. PAPER
Ann. of Probability 34, 1370-1422 (2006)
ABSTRACT:
El. J. of Probability 11, 460-485 (2006)
ABSTRACT:
We consider the weak coupling limit for a quantum system consisting of a small subsystem and reservoirs. It is known rigorously since the work of E.B.Davies that the Heisenberg evolution restricted to the small system converges in an appropriate sense to a Markovian semigroup. In the nineties, Accardi, Frigerio and Lu initiated an investigation of the convergence of the unreduced unitary evolution to a singular unitary evolution generated by a quantum Langevin equation. We present a version of this convergence which is both simpler and stronger than the formulations which we know. Our main result says that in an appropriately understood weak coupling limit the interaction of the small system with environment can be expressed in terms of the so-called quantum white noise. PAPER
ABSTRACT: The $m\,\alpha^6$ correction to energy is expressed in terms of an effective Hamiltonian $H^{(6)}$ for an arbitrary state of helium. Numerical calculations are performed for $n=2$ levels and the previous result for $2^3P$ centroid is corrected. While the resulting theoretical predictions for the ionization energy are in a moderate agreement with experimental values for $2^3S_1$, $2^3P$, and $2^1S_0$ states, they are in a significant disagreement for the singlet state $2^1P_1$.
ABSTRACT: Mean field quantum random graphs give a natural generalization of classical Erd\H{o}s-R\'{e}nyi percolation model on complete graph $G_N$ with $p =\beta /N$. Quantum case incorporates an additional parameter $\lambda\geq 0$, and the short-long range order transition should be studied in the $(\beta ,\lambda)$-quarter plane. In this work we explicitly compute the corresponding critical curve $\gamma_c$, and derive results on two-point functions and sizes of connected components in both short and long range order regions. In this way the classical case corresponds to the limiting point $(\beta_c ,0) = (1,0)$ on $\gamma_c$.
ABSTRACT: We consider two-dimensional Schr\"odinger operators in bounded domains. We analyze relations between the nodal domains, spectral minimal partitions and spectral properties of the corresponding operator. The main results concern the existence and regularity of the minimal partitions and the characterization of the minimal partitions associated with nodal sets as the nodal domains of Courant--sharp eigenfunctions.
Lecture Notes in Physics 695
Abstract: Large Coulomb Systems explores a selection of mathematical topics inspired by QED. Most of its contributions is based on lectures from the Summer School in Nordfjordeid, Norway, organized by the Network Analysis and Quantum.
Commun. Math. Phys. Vol. 268, 621--672 (2006)
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ESI News Vol. 1, 4--7, (2006)
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accepted by Phys. Rev. B
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Abstract: We first prove a weighted inequality of Moser-Trudinger type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than $-1$. Without symmetry assumption, it holds if and only if the parameter is in the interval $(-1,0]$. The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Hardy-Sobolev inequality, as established by Caffarelli-Kohn-Nirenberg, in two space dimensions. In fact, for suitable sets of parameters (asymptotically sharp) we prove symmetry or symmetry breaking by means of a blow-up method. In this way, the weighted Moser-Trudinger inequality appears as a limit case of the Hardy-Sobolev inequality.