Zeta-regularisation for exact-WKB resolution of a general 1D Schr\"odinger equation

TL;DR

This paper presents an exact WKB-based method using zeta-regularisation for solving 1D Schrödinger equations with polynomial potentials, providing precise quantisation conditions and spectral information.

Contribution

It introduces a novel exact quantisation approach combining spectral and classical zeta-regularisations, extending Bethe-Ansatz formulas for 1D quantum systems.

Findings

Derivation of exact Bohr–Sommerfeld-like quantisation conditions

Ability to evaluate spectral determinants and wavefunctions

Extension of integrable system Bethe-Ansatz formulas

Abstract

We review an exact analytical resolution method for general one-dimensional (1D) quantal anharmonic oscillators: stationary Schr\"odinger equations with polynomial potentials. It is an exact form of WKB treatment involving spectral (usual) vs "classical" (newer) zeta-regularisations in parallel. The central results are a set of Bohr--Sommerfeld-like but exact quantisation conditions, directly drawn from Wronskian identities, and appearing to extend Bethe-Ansatz formulae of integrable systems. Such exact quantisation conditions do not just select the eigenvalues; some evaluate the spectral determinants, and others the wavefunctions, for the spectral parameter in general position.