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

Abstract

AbstractWe consider the operator urn:x-wiley:00222526:media:sapm12367:sapm12367-math-0001subject to the Dirichlet or Robin condition, where a domain is bounded or unbounded. The symbol stands for a second‐order self‐adjoint differential operator on such that the spectrum of the operator contains several discrete eigenvalues , . These eigenvalues are thresholds in the essential spectrum of the operator . We study how these thresholds bifurcate once we add a small localized perturbation to the operator , where is a small positive parameter and is an abstract, not necessarily symmetric operator. We show that these thresholds bifurcate into eigenvalues and resonances of the operator in the vicinity of for sufficiently small . 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 () 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 .