Asymptotic Expansion of the Eigenvalues of a Bathtub Potential with Quadratic Ends

TL;DR

This paper derives an asymptotic expansion for eigenvalues of a semiclassical Schrödinger operator with a potential having quadratic ends and a possible flat region, revealing oscillatory behavior and implications for trace formulas.

Contribution

It introduces a novel asymptotic expansion for eigenvalues of non-smooth potentials with quadratic ends, extending semiclassical analysis techniques.

Findings

Asymptotic expansion valid in high energy or semiclassical regimes

Leading order given by Bohr-Sommerfeld quantization

Oscillatory coefficients in the expansion

Abstract

We consider the eigenvalues of a one-dimensional semiclassical Schr\"odinger operator, where the potential consist of two quadratic ends (that is, looks like a harmonic oscillator at each infinite end), possibly with a flat region in the middle. Such a potential notably has a discontinuity in the second derivative. We derive an asymptotic expansion, valid either in the high energy regime or the semiclassical regime, with a leading order term given by the Bohr-Sommerfeld quantization condition, and an asymptotic expansion consisting of negative powers of the leading order term, with coefficients that are oscillatory in the leading order term. We apply this expansion to study the results of the Gutzwiller Trace formula and the heat kernel asymptotic for this class of potentials, giving an idea into what results to expect for such trace formulas for non-smooth potentials.