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

Result code in 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>

Alternative codes found

RIV/62690094:18470/21:50017892

Result on the web

<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>

Result language

angličtina

Original language name

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

Original language description

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.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

—

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2021

Confidentiality

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

Name of the periodical

Studies in Applied Mathematics

ISSN

0022-2526

e-ISSN

1467-9590

Volume of the periodical

146

Issue of the periodical within the volume

4

Country of publishing house

US - UNITED STATES

Number of pages

47

Pages from-to

834-880

UT code for WoS article

000608177800001

EID of the result in the Scopus database

2-s2.0-85099380474