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

TL;DR

This paper extends previous results on the spectral properties of Schr"odinger operators with polynomial potentials, proving the connectedness and irreducibility of the spectral determinant for arbitrary degrees and analyzing parameter space structures.

Contribution

It generalizes the irreducibility result of the spectral discriminant to polynomial potentials of any degree and studies the topology of parameter spaces for eigenfunctions with multiple boundary conditions.

Findings

Spectral determinant is connected and irreducible for arbitrary degree polynomial potentials.

Parameter spaces for eigenfunctions with multiple boundary conditions are connected, except in specific even-degree cases.

Connected components are distinguished by the number of zeros of eigenfunctions in certain cases.

Abstract

In this paper, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schr\"odinger equation with quartic potentials. We consider the eigenvalue problem with a complex-valued polynomial potential of arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation. We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k>2 boundary conditions, except for the case d is even and k=d/2. In the latter case, connected components of the parameter space are distinguished by the number of zeros of the eigenfunctions.