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

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