TBA equations for the Schr\"odinger equation with a regular singularity

TL;DR

This paper develops Thermodynamic Bethe Ansatz equations for Schrödinger equations with polynomial potentials and singularities, extending the ODE/IM correspondence and solving related Riemann-Hilbert problems, with applications to spectral computations.

Contribution

It introduces TBA equations for Schrödinger equations with singularities, generalizing existing methods and providing new tools for spectral analysis.

Findings

Derived TBA equations for polynomial potentials with singularities.

Connected TBA equations to the Riemann-Hilbert problem in WKB analysis.

Numerically computed Voros spectra for specific potentials.

Abstract

We derive the Thermodynamic Bethe Ansatz (TBA) equations for the Schr\"odinger equation with an arbitrary polynomial potential and a regular singular (simple and double pole) term. The TBA equations provide a non-trivial generalization of the ODE/IM correspondence and also give a solution for the Riemann-Hilbert problem in the exact WKB method. We study the TBA equations in detail for the linear and the harmonic oscillator potentials together with inverse and centrifugal terms. As an application, we also compute numerically the Voros spectrum for these potentials using the Bohr-Sommerfeld quantization condition.