Trace formulas for the modified Mathieu equation

Abstract

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