On eigenvalues of the Schr\"odinger operator with an even complex-valued polynomial potential

TL;DR

This paper extends previous work on eigenvalues of Schrödinger operators by analyzing complex polynomial potentials of arbitrary degree, revealing the structure of spectral determinants and the connectedness of parameter spaces.

Contribution

It generalizes earlier results to arbitrary polynomial degrees and boundary conditions, detailing the spectral determinant's components and the topology of parameter spaces.

Findings

Spectral determinant has two components for even and odd eigenvalues.

Parameter space with maximal boundary conditions has infinitely many connected components.

Results extend understanding of eigenvalues for complex polynomial potentials.

Abstract

In this paper, we generalize several results of the article "Analytic continuation of eigenvalues of a quartic oscillator" of A. Eremenko and A. Gabrielov. We consider a family of eigenvalue problems for a Schr\"odinger equation with even polynomial potentials of arbitrary degree d with complex coefficients, and k<(d+2)/2 boundary conditions. We show that the spectral determinant in this case consists of two components, containing even and odd eigenvalues respectively. In the case with k=(d+2)/2 boundary conditions, we show that the corresponding parameter space consists of infinitely many connected components.