TBA Equations and Quantization Conditions

TL;DR

This paper explores the use of TBA equations derived from the exact WKB method to solve spectral problems in quantum mechanics, including computing quantum corrections and resonances for various potentials.

Contribution

It develops a graphical procedure for analyzing wall-crossing in TBA equations and applies it to compute exact spectra and quantum corrections in multiple examples.

Findings

Successfully computes quantum corrections to WKB periods.

Determines exact spectra for various polynomial potentials.

Identifies resonances in unbounded potentials.

Abstract

It has been recently realized that, in the case of polynomial potentials, the exact WKB method can be reformulated in terms of a system of TBA equations. In this paper we study this method in various examples. We develop a graphical procedure due to Toledo, which provides a fast and simple way to study the wall-crossing behavior of the TBA equations. When complemented with exact quantization conditions, the TBA equations can be used to solve spectral problems exactly in Quantum Mechanics. We compute the quantum corrections to the all-order WKB periods in many examples, as well as the exact spectrum for many potentials. In particular, we show how this method can be used to determine resonances in unbounded potentials.