Trace formulas for the modified Mathieu equation

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{d}\mathrm{e}\mathrm{t}$_{F}(H-\lambda I)$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.