On eigenvalues of the Schrödinger operator with an even complex-valued polynomial potential

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 Schrödinger 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.

References in corpus (4)

• Analytic continuation of eigenvalues of a quartic oscillator
• On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential
• Singular perturbation of polynomial potentials in the complex domain with applications to PT-symmetric families
• Irreducibility of some spectral determinants

Cited by in corpus (2)

• On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential
• Explicit energy expansion for general odd degree polynomial potentials