On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity

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Abstract

We consider the Schrödinger operator on the plane with bounded potential, where is a real potential, and are compactly supported complex potentials, and 
 is a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator consists of a pair of isolated eigenvalues and the essential spectrum of the operator has a virtual level at its lower edge and a spectral singularity inside.

Additionally, we assume that there is a certain superposition of eigenvalues of the operator with the virtual level and spectral singularity of the operator; this leads to the emergence of a special threshold in the essential spectrum of the perturbed operator, with the perturbation leading to a bifurcation of this threshold into eigenvalues and resonances with multiplicity doubling. The bifurcation scenario described in this paper is qualitatively different from the previously known ones.

About the authors

D. I Borisov

Institute of Mathematics with Computing Centre, Subdivision of the Ufa Federal Research Centre, Russian Academy of Sciences, Ufa, 450008, Russia; Akmulla Bashkir State Pedagogical University, Ufa, 450008, Russia; Univerzita Hradec Králové, Hradec Králové III, 500 03, Czech Republic

Email: borisovdi@yandex.ru
г. Уфа, Росия;г. Градец Кралове, Чехия

D. A Zezyulin

ITMO University, St. Petersburg, 197101, Russia

Author for correspondence.
Email: d.zezyulin@gmail.com
г. Санкт-Петербург, Россия

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