Scaling properties of the pairing problem in the strong coupling limit

TL;DR

This paper develops an expansion for the excited states of the pairing Hamiltonian in the strong coupling limit, demonstrating that a few statistical moments suffice for accurate energy estimates and analyzing the convergence properties of the series.

Contribution

It provides an analytic expansion for excited state energies in the strong coupling regime and clarifies the relationship between convergence radius and level distribution.

Findings

Few statistical moments suffice for accurate energy estimates at moderate coupling.

Analytic expressions for the first four terms of the energy series are derived.

Convergence radius depends on level distribution, not critical coupling points.

Abstract

We study the excited states of the pairing Hamiltonian providing an expansion for their energy in the strong coupling limit. To assess the role of the pairing interaction we apply the formalism to the case of a heavy atomic nucleus. We show that only a few statistical moments of the level distribution are sufficient to yield an accurate estimate of the energy for not too small values of the coupling $G$ and we give the analytic expressions of the first four terms of the series. Further, we discuss the convergence radius $G_{\rm sing}$ of the expansion showing that it strongly depends upon the details of the level distribution. Furthermore $G_{\rm sing}$ is not related to the critical values of the coupling $G_{\rm crit}$, which characterize the physics of the pairing Hamiltonian, since it can exist even in the absence of these critical points.