Beyond the WKB approximation in PT-symmetric quantum mechanics

TL;DR

This paper develops a systematic approximation scheme for PT-symmetric quantum mechanics beyond WKB, revealing the importance of nonperturbative effects and connecting to integrable quantum field theories.

Contribution

It introduces a novel approach using the ODE/IM correspondence to analyze energy level mergings and symmetry breaking in PT-symmetric models, including centrifugal terms.

Findings

The scheme captures all features of symmetry-breaking transitions.

Nonperturbative effects dominate level behavior post-transition.

New method recovers conserved charges without resummation.

Abstract

The mergings of energy levels associated with the breaking of PT symmetry in the model of Bender and Boettcher, and in its generalisation to incorporate a centrifugal term, are analysed in detail. Even though conventional WKB techniques fail, it is shown how the ODE/IM correspondence can be used to obtain a systematic approximation scheme which captures all previously-observed features. Nonperturbative effects turn out to play a crucial role, governing the behaviour of almost all levels once the symmetry-breaking transition has been passed. In addition, a novel treatment of the radial Schrodinger equation is used to recover the values of local and non-local conserved charges in the related integrable quantum field theories, without any need for resummation even when the angular momentum is nonzero.