On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity
- Authors: Borisov D.I1,2,3, Zezyulin D.A4
-
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://journals.rcsi.science/0374-0641/article/view/144919
- DOI: https://doi.org/10.31857/S0374064123020127
- EDN: https://elibrary.ru/PVRLHB
- ID: 144919
Cite item
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
References
- Guseinov G.Sh. On the concept of spectral singularities // Pramana - J. Phys. 2009. V. 73. № 3. P. 587-603.
- Borisov D.I., Zezyulin D.A., Znojil M. Bifurcations of thresholds in essential spectra of elliptic operators under localized non-Hermitian perturbations // Stud. Appl. Math. 2021. V. 146. № 4. P. 834-880.
- Borisov D.I., Zezyulin D.A. Bifurcations of essential spectra generated by a small non-Hermitian hole. I. Meromorphic continuations // Russ. J. Math. Phys. 2021. V. 28. № 4. P. 416-433.
- Borisov D.I., Zezyulin D.A. Bifurcations of essential spectra generated by a small non-Hermitian small hole. II. Eigenvalues and resonances // Russ. J. Math. Phys. 2022. V. 29. № 3. P. 321-341.
- Назаров С.А. Сохранение пороговых резонансов и отцепление собственных чисел от порога непрерывного спектра квантовых волноводов // Мат. сб. 2021. Т. 212. № 7. С. 84-121.
- Назаров С.А. Пороговые резонансы и виртуальные уровни в спектре цилиндрических и периодических волноводов // Изв. РАН. Сер. мат. 2020. Т. 84. № 6. С. 73-130.
- Гатауллин Т.М., Карасёв М.В. О возмущении квазиуровней оператора Шрёдингера с комплексным потенциалом // Теор. мат. физ. 1971. Т. 9. № 2. С. 252-263.
- Лакаев С.Н., Абдухакимов С.Х. Пороговые эффекты в системе двух фермионов на оптической решётке // Теор. мат. физ. 2020. Т. 203. № 2. С. 251-268.
- Лакаев С.Н., Улашов С.С. Существование и аналитичность связанных состояний двухчастичного оператора Шрёдингера на решётке // Теор. мат. физ. 2012. Т. 170. № 3. С. 393-408.
- Gesztesy F., Holden H. A unified approach to eigenvalues and resonances of Schr"odinger operators using Fredholm determinants // J. Math. Anal. Appl. 1987. V. 123. № 1. P. 181-198.
- Борисов Д.И. Возмущение края существенного спектра волновода с окном. I. Убывающие резонансные решения // Пробл. мат. анализа. 2014. Т. 77. С. 19-54.
- Borisov D.I., Zezyulin D.A. Sequences of closely spaced resonances and eigenvalues for bipartite complex potentials // Appl. Math. Lett. 2020. V. 100. ID 106049.
- Klopp F. Resonances for large one-dimensional "ergodic" systems // Analysis and PDE. 2016. V. 9. № 2. P. 259-352.
![](/img/style/loading.gif)