Exact quantization and analytic continuation

TL;DR

This paper derives an exact quantization condition for one-dimensional quantum systems with polynomial potentials, extends it to singular potentials, and explores the analytic continuation of spectra using TBA equations and numerical checks.

Contribution

It provides a streamlined derivation of the EQC, generalizes it to singular potentials, and investigates spectral monodromies via TBA equations and numerical validation.

Findings

Derived the exact quantization condition using Wronskian relations.

Extended the EQC to potentials with regular singularities.

Numerically verified the TBA equations and explored spectral continuation.

Abstract

In this paper we give a streamlined derivation of the exact quantization condition (EQC) on the quantum periods of the Schr\"odinger problem in one dimension with a general polynomial potential, based on Wronskian relations. We further generalize the EQC to potentials with a regular singularity, describing spherical symmetric quantum mechanical systems in a given angular momentum sector. We show that the thermodynamic Bethe ansatz (TBA) equations that govern the quantum periods undergo nontrivial monodromies as the angular momentum is analytically continued between integer values in the complex plane. The TBA equations together with the EQC are checked numerically against Hamiltonian truncation at real angular momenta and couplings, and are used to explore the analytic continuation of the spectrum on the complex angular momentum plane in examples.