Winter (or delta-shell) Model at Small and Intermediate Volumes

TL;DR

This paper analyzes the Winter (delta-shell) model at finite volume, exploring how resonance spectra emerge as the system transitions from simple oscillations to resonant states, and develops new high-energy expansion methods for this quantum system.

Contribution

It introduces novel high-energy expansions that resum perturbative series for the Winter model at finite volume, valid across a range of energies and couplings, with two complementary resummation schemes.

Findings

Resonant states form clusters at finite volume.

New high-energy expansions converge quickly and are valid at low energies.

Two resummation schemes provide consistent approximations with different physical insights.

Abstract

We consider Winter (or delta-shell) model at finite volume, describing a small resonating cavity weakly coupled to a large one, for small and intermediate volumes (lengths). By defining N as the ratio of the length of the large cavity over the small one, we study the symmetric case N=1, in which the two cavities actually have the same length, as well as the cases N=2,3,4. By increasing N in the above range, the transition from a simple quantum oscillating system to a system having a resonance spectrum is investigated. We find that each resonant state is represented, at finite volume, by a cluster of states, each one resonating in a specific coupling region, centered around a state resonating at very small couplings. We derive high-energy expansions for the particle momenta in the above N cases, which (approximately) resum their perturbative series to all orders in the coupling among the cavities. These new expansions converge rather quickly with the order, provide, surprisingly, a uniform approximation in the coupling and also work, again surprisingly, at low energies. We construct a first resummation scheme having a clear physical picture, which is based on a function-series expansion, as well as a second scheme based on a recursion equation. The two schemes coincide at leading order, while they differ from next-to-leading order on. In particular, the recursive scheme realizes an approximate resummation of the function-series expansion generated within the first scheme.