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

Abstract

In this paper, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schrödinger 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.

23 pages

References in corpus (5)

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

Cited by in corpus (6)

• Quasi-exactly solvable quartic: elementary integrals and asymptotics
• On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential
• Level Crossing in Random Matrices: I. Random perturbation of a fixed matrix
• On eigenvalues of the Schrödinger operator with an even complex-valued polynomial potential
• Explicit energy expansion for general odd degree polynomial potentials
• Quasi-exactly solvable quartic: real algebraic spectral locus