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

Abstract

We consider the operator $$\mathcal{H}={\mathcal{H}}^{\prime }-\frac{{\partial }^2}{\partial x_d^2}\text{on}\omega \times \mathbb{R}$$ subject to the Dirichlet or Robin condition, where a domain $\omega \subseteq {\mathbb{R}}^{d-1}$ is bounded or unbounded. The symbol ${\mathcal{H}}^{\prime }$ stands for a second order self-adjoint differential operator on $\omega$ such that the spectrum of the operator ${\mathcal{H}}^{\prime }$ contains several discrete eigenvalues ${\Lambda }_j$, $j=1,\dots ,m$. These eigenvalues are thresholds in the essential spectrum of the operator $\mathcal{H}$. We study how these thresholds bifurcate once we add a small localized perturbation $\epsilon \mathcal{L}(\epsilon )$ to the operator $\mathcal{H}$, where $\epsilon$ is a small positive parameter and $\mathcal{L}(\epsilon )$ is an abstract, not necessarily symmetric operator. We show that these thresholds bifurcate into eigenvalues and resonances of the operator $\mathcal{H}$ 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 non-self-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. We use 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.

37 pages, 2 figures; several corrections made; accepted for Studies in Applied Mathematics

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