Singular perturbation of polynomial potentials in the complex domain with applications to PT-symmetric families

Abstract

In the first part of the paper, we discuss eigenvalue problems of the form -w"+Pw=Ew with complex potential P and zero boundary conditions at infinity on two rays in the complex plane. We give sufficient conditions for continuity of the spectrum when the leading coefficient of P tends to 0. In the second part, we apply these results to the study of topology and geometry of the real spectral loci of PT-symmetric families with P of degree 3 and 4, and prove several related results on the location of zeros of their eigenfunctions.

The main result on singular perturbation is substantially improved, generalized, and the proof is simplified. 37 pages, 16 figures

References in corpus (2)

• Analytic continuation of eigenvalues of a quartic oscillator
• Irreducibility of some spectral determinants

Cited by in corpus (5)

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• On eigenvalues of the Schrödinger operator with an even complex-valued polynomial potential
• A Note on EVgrafov-Fedoryuk's theory and quadratic differentials