Bifurcations of thresholds in essential spectra of elliptic operators under localized non-Hermitian perturbations

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61389005%3A_____%2F21%3A00539479" target="_blank" >RIV/61389005:_____/21:00539479 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/62690094:18470/21:50017892

Výsledek na webu

<a href="https://doi.org/10.1111/sapm.12367" target="_blank" >https://doi.org/10.1111/sapm.12367</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1111/sapm.12367" target="_blank" >10.1111/sapm.12367</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Bifurcations of thresholds in essential spectra of elliptic operators under localized non-Hermitian perturbations

Popis výsledku v původním jazyce

We consider the operatornnH = H'- partial derivative(2)/partial derivative x(d)(2) on omega x Rnnsubject to the Dirichlet or Robin condition, where a domain omega subset of Rd-1 is bounded or unbounded. The symbol H ' stands for a second-order self-adjoint differential operator on omega such that the spectrum of the operator H ' contains several discrete eigenvalues Lambda j, j=1, ... ,m. These eigenvalues are thresholds in the essential spectrum of the operator H. We study how these thresholds bifurcate once we add a small localized perturbation epsilon L(epsilon) to the operator H, where epsilon is a small positive parameter and L(epsilon) is an abstract, not necessarily symmetric operator. We show that these thresholds bifurcate into eigenvalues and resonances of the operator H in the vicinity of Lambda(j) for sufficiently small epsilon. We prove effective simple conditions determining the existence of these resonances and eigenvalues and find the leading terms of their asymptotic expansions. Our analysis applies to generic nonself-adjoint perturbations and, in particular, to perturbations characterized by the parity-time (PT) symmetry. Potential applications of our result embrace a broad class of physical systems governed by dispersive or diffractive effects. As a case example, we employ our findings to develop a scheme for a controllable generation of non-Hermitian optical states with normalizable power and real part of the complex-valued propagation constant lying in the continuum. The corresponding eigenfunctions can be interpreted as an optical generalization of bound states embedded in the continuum. For a particular example, the persistence of asymptotic expansions is confirmed with direct numerical evaluation of the perturbed spectrum.

Název v anglickém jazyce

Bifurcations of thresholds in essential spectra of elliptic operators under localized non-Hermitian perturbations

Popis výsledku anglicky

We consider the operatornnH = H'- partial derivative(2)/partial derivative x(d)(2) on omega x Rnnsubject to the Dirichlet or Robin condition, where a domain omega subset of Rd-1 is bounded or unbounded. The symbol H ' stands for a second-order self-adjoint differential operator on omega such that the spectrum of the operator H ' contains several discrete eigenvalues Lambda j, j=1, ... ,m. These eigenvalues are thresholds in the essential spectrum of the operator H. We study how these thresholds bifurcate once we add a small localized perturbation epsilon L(epsilon) to the operator H, where epsilon is a small positive parameter and L(epsilon) is an abstract, not necessarily symmetric operator. We show that these thresholds bifurcate into eigenvalues and resonances of the operator H in the vicinity of Lambda(j) for sufficiently small epsilon. We prove effective simple conditions determining the existence of these resonances and eigenvalues and find the leading terms of their asymptotic expansions. Our analysis applies to generic nonself-adjoint perturbations and, in particular, to perturbations characterized by the parity-time (PT) symmetry. Potential applications of our result embrace a broad class of physical systems governed by dispersive or diffractive effects. As a case example, we employ our findings to develop a scheme for a controllable generation of non-Hermitian optical states with normalizable power and real part of the complex-valued propagation constant lying in the continuum. The corresponding eigenfunctions can be interpreted as an optical generalization of bound states embedded in the continuum. For a particular example, the persistence of asymptotic expansions is confirmed with direct numerical evaluation of the perturbed spectrum.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

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

Název periodika

Studies in Applied Mathematics

ISSN

0022-2526

e-ISSN

1467-9590

Svazek periodika

146

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

47

Strana od-do

834-880

Kód UT WoS článku

000608177800001

EID výsledku v databázi Scopus

2-s2.0-85099380474