Equidistant-level model (infinitely many levels)

Let us consider a phase space of infinitely many, doubly-degenerate,
equidistant single-particle levels spaced by *d*. Suppose
that all the levels up to a certain Fermi energy
are occupied by the fermions, and that they
interact through a two-body interaction .
The Hamiltonian of the model reads,

where is the number operator of the

where

Of course, we do not intend to consider here infinite numbers of
particles and infinite energies, cf. Eq. (49). In practice,
we should assume
that the number of fermions interacting through
is finite but large.
The remaining ones (e.g., occupying the most bound shells) form an
inert core. This guarantees the effect of the finite
spectrum not to be important. Consequently, in future discussions
we assume that *N*1 and that the first level belonging to the
``interaction-active" space has *k*=1.

Irrespective of how complicated interaction
is, its
assumed shift-symmetry allows for an exact analysis in terms of
filters based on ground-state energy differences. Indeed, for any
particle number *N*, the ground-state energy reads

where

From
the shift-symmetry (50), it is obvious that the change in the
ground-state energy, occurring when the interaction is switched on, is
the same for all even systems (
)
and the same for all odd systems
(
)
and
hence

where is given by Eq. (2).

Filters (1), (6), and (7) now give

hence

It is quite remarkable that the above argumentation does not at all depend on details of the two-body interaction, and that - irrespective of the interaction - filters (6) and (7) correctly separate the interaction effects (54) from the single-particle spacings (55).

This generic result does not depend on whether any approximations are used to obtain the ground-state energies of interacting systems. In particular, the BCS mean-field results (obtained for interaction ) will also obey the pattern presented in Eqs. (54) exactly; only the value of the interaction-energy difference may be different from the exact result.

Similarly, the BCS results obtained for the
seniority-pairing interaction (14) also follow the same
pattern; however, in this case we have to additionally ensure that
the phase space in which the pairing correlations are allowed to
develop is the same for all particle numbers. Indeed,
one usually solves the BCS equations in a
finite phase-space window, adjusting the interaction strength *G* to
that window. For the results (54) to be valid, we have
to always use the same number of levels below and above the Fermi
level, independently of the number of particles. Actually, such a
prescription is often used in realistic BCS or HFB nuclear
structure calculations.

As discussed above, in the case of the equidistant-level model,
the energy-filter
reproduces the value of the level spacing *d*regardless of the detailed structure of
and details of
the many-body approximation used. The reason for this is the shift-symmetry
of the Hamiltonian. Of course, if this symmetry is broken either by
assuming the finite Hilbert space (see the following section)
or by introducing the explicit
symmetry-violating terms, one cannot *a priori* expect the
above conclusion to hold *exactly*.

It is also worth noting that the OES (54) is defined in terms of the pairing (interaction) energy in the even and odd system, and not in terms of the HFB or BCS pairing gap (defined as half of the lowest excitation energy). Only for strong pairing correlations are the pairing gaps and the values of the OES the same (see Ref. [11]). In nuclei, pairing correlations never reach such a limit, and the OES and BCS pairing gap in an odd system can be significantly different.