Accuracy of solutions

Benchmark tests of the HFB part of our calculations are reported in Ref. [72]. Since the accuracy of the canonical wave functions, in which the QRPA calculations are carried out, strongly affects the quality of results (in particular QRPA self consistency), we take special care to compute them precisely. As discussed in Sec. 2, we obtain canonical states by diagonalizing the single-particle density matrix $\rho$ represented in the orthonormalized set of functions $\{ \varphi^{\mu}_1({\mbox{\boldmath$r$\unboldmath }}) +\varphi^{\mu}_2( {\mbox{\boldmath$r$\unboldmath }})\}$. The accuracy of this method is illustrated in Fig. 1, which plots the quasiparticle energies $E_\mu ^{\rm check}$, obtained by diagonalizing the HFB Hamiltonian in the canonical basis (Eq. (4.20) of Ref. [60]), versus the quasiparticle energies $E_\mu$ obtained by solving the HFB differential equations directly in coordinate space (Eq. (4.10) of Ref. [60]). Two sets of canonical states are used: (i) those obtained through the procedure outlined above (dotted line) and (ii) those obtained in the standard way by diagonalizing the density matrix $\rho(\mbox{\boldmath$r$\unboldmath },\mbox{\boldmath$r$\unboldmath }^\prime)$ in discretized coordinate space (Eq. (3.24a) of Ref. [60]; solid line).

Figure 1: Neutron quasiparticle energies $E_\mu^{\rm check}$ for $s_{1/2}$ states in $^{174}$Sn, calculated by diagonalizing the HFB Hamiltonian in the canonical basis, versus the quasiparticle energies $E_\mu$ obtained by directly solving the HFB equations in coordinate space. Standard (solid line, diamonds) and improved (dotted line, dots) methods are used to obtain the canonical states. See text for details.

If the canonical basis is precisely determined, $E_\mu ^{\rm check}$=$E_\mu$ and the two sets of $E_\mu ^{\rm check}$ coincide. Within the standard approach, however, the high canonical energies deviate visibly from their HFB counterparts, i.e., the accuracy of the underlying canonical wave functions is poor. On the other hand, the quasiparticle energies and canonical wave functions calculated within the modified approach introduced above are as accurate as the original solutions to the HFB equations, even for high-lying nearly-empty states. (See also Sec. VI.D of Ref. [73] for a discussion relevant to this point.)

Having examined the canonical basis, we turn to the accuracy of the QRPA part of the calculation. To test it, we first consider solutions related to symmetries. If a Hamiltonian is invariant under a symmetry operator $\hat{P}$ and the HFB state $\vert\Psi\rangle$ spontaneously breaks the symmetry, then $e^{i\alpha\hat{P}}\vert\Psi\rangle$, with $\alpha$ an arbitrary c-number, is degenerate with the state $\vert\Psi\rangle$. The QRPA equations have a spurious solution at zero energy associated with the symmetry breaking [8,74], while all other solutions are free of the spurious motion. This property is important for strength functions, and gives us a way of testing the calculations. Since our QRPA equations, which assume spherical symmetry, are based on mean fields that include pairing and are localized in space, there appear spurious states associated with particle-number nonconservation (proton and/or neutron; 0$^+$ channel) and center-of-mass motion (1$^-$ channel). These two cases are discussed below in Sec. 3.1 and Sec. 3.2.