On Perturbation of Thresholds in Essential Spectrum under Coexistence of Virtual Level and Spectral Singularity

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

On Perturbation of Thresholds in Essential Spectrum under Coexistence of Virtual Level and Spectral Singularity

Popis výsledku v původním jazyce

We study the perturbation of the Schr &amp; 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 &amp; 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 &amp; 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 &gt; 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 &amp; ouml;dinger operator is described.

Název v anglickém jazyce

On Perturbation of Thresholds in Essential Spectrum under Coexistence of Virtual Level and Spectral Singularity

Popis výsledku anglicky

We study the perturbation of the Schr &amp; 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 &amp; 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 &amp; 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 &gt; 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 &amp; ouml;dinger operator is described.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Russian journal of mathematical physics

ISSN

1061-9208

e-ISSN

1555-6638

Svazek periodika

31

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

19

Strana od-do

60-78

Kód UT WoS článku

001186903600002

EID výsledku v databázi Scopus

2-s2.0-85188064518