On Perturbation of Thresholds in Essential Spectrum under Coexistence of Virtual Level and Spectral Singularity
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F24%3A50021853" target="_blank" >RIV/62690094:18470/24:50021853 - isvavai.cz</a>
Result on the web
<a href="https://link.springer.com/article/10.1134/S106192084010059" target="_blank" >https://link.springer.com/article/10.1134/S106192084010059</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1134/S106192084010059" target="_blank" >10.1134/S106192084010059</a>
Alternative languages
Result language
angličtina
Original language name
On Perturbation of Thresholds in Essential Spectrum under Coexistence of Virtual Level and Spectral Singularity
Original language description
We study the perturbation of the Schr & ouml;dinger operator on the plane with a bounded potential of the form V-1(x)+V-2(y), where V-1 is a real function and V-2 is a compactly supported function. It is assumed that the one-dimensional Schr & ouml;dinger operator H-1 with the potential V-1 has two real isolated eigenvalues Lambda(0), Lambda(1) in the lower part of its spectrum, and the one-dimensional Schr & ouml;dinger operator H-2 with the potential V-2 has a virtual level at the boundary of its essential spectrum, i.e., at lambda = 0, and a spectral singularity at the inner point of the essential spectrum lambda = mu > 0. In addition, the eigenvalues and the spectral singularity overlap in the sense of the equality lambda(0 ): = Lambda(0 )+ mu = Lambda(1). We show that a perturbation by an abstract localized operator leads to a bifurcation of the internal threshold lambda(0) into four spectral objects which are resonances and/or eigenvalues. These objects correspond to the poles of the local meromorphic continuations of the resolvent. The spectral singularity of the operator H-2 qualitatively changes the structure of these poles as compared to the previously studied case where no spectral singularity was present. This effect is examined in detail, and the asymptotic behavior of the emerging poles and corresponding spectral objects of the perturbed Schr & ouml;dinger operator is described.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10102 - Applied mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Russian journal of mathematical physics
ISSN
1061-9208
e-ISSN
1555-6638
Volume of the periodical
31
Issue of the periodical within the volume
1
Country of publishing house
US - UNITED STATES
Number of pages
19
Pages from-to
60-78
UT code for WoS article
001186903600002
EID of the result in the Scopus database
2-s2.0-85188064518