Trace formulas for the modified Mathieu equation

TL;DR

This paper develops a method to derive trace formulas for Schrödinger operators with growing potentials, exemplified by the modified Mathieu equation, providing asymptotic expansions of the Fredholm determinant as the spectral parameter tends to negative infinity.

Contribution

It introduces a novel approach to obtain trace identities for Schrödinger operators with unbounded potentials, specifically applied to the modified Mathieu equation.

Findings

Derived asymptotic expansion of the Fredholm determinant for the modified Mathieu operator.

Established trace formulas for Schrödinger operators with growing potentials.

Illustrated the method with the example of the modified Mathieu equation.

Abstract

For the radial and one-dimensional Schr\"{o}dinger operator $H$ with growing potential $q(x)$ we outline a method of obtaining the trace identities - an asymptotic expansion of the Fredholm determinant $\mathrm{det}_{F}(H-\lambda I)$ as $\lambda\to-\infty$. As an illustrating example, we consider Schr\"{o}dinger operator with the potential $q(x)=2\cosh 2x$, associated with the modified Mathieu equation.