Airy function (exact WKB results for potentials of odd degree)

TL;DR

This paper extends the exact WKB analysis of 1D Schrödinger operators to odd-degree polynomial potentials, revealing new dualities and identities involving Airy functions and spectral determinants, with applications to cubic and linear potentials.

Contribution

It introduces a novel extension of the exact WKB method to odd-degree potentials, uncovering dualities and classical identities in a unified framework.

Findings

Duality between linear and quartic potentials involving Airy functions

New identities for Airy functions derived from spectral determinants

Numerical validation of the extended WKB formalism for odd potentials

Abstract

An exact WKB treatment of 1-d homogeneous Schr\"odinger operators (with the confining potentials $q^N$, $N$ even) is extended to odd degrees $N$. The resulting formalism is first illustrated theoretically and numerically upon the spectrum of the cubic oscillator (potential $|q|^3$). Concerning the linear potential (N=1), the theory exhibits a duality in which the Airy functions Ai, Ai' become paired with the spectral determinants of the quartic oscillator (N=4). Classic identities for the Airy function, as well as some less familiar ones, appear in this new perspective as special cases in a general setting.