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On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity
- Authors: Borisov D.I1,2,3, Zezyulin D.A4
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Affiliations:
- 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
- ITMO University, St. Petersburg, 197101, Russia
- Issue: Vol 59, No 2 (2023)
- Pages: 270-274
- Section: Articles
- URL: https://journal-vniispk.ru/0374-0641/article/view/144919
- DOI: https://doi.org/10.31857/S0374064123020127
- EDN: https://elibrary.ru/PVRLHB
- ID: 144919
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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
D. A Zezyulin
ITMO University, St. Petersburg, 197101, Russia
Author for correspondence.
Email: d.zezyulin@gmail.com
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