On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity

Abstract

We consider the Schrödinger operator on the plane with bounded potential $V_1(x)\!+\!V_2(y)\!+\!\varepsilon W(x,y)$, where $V_1$ is a real potential, $V_2$ and $W$ are compactly supported complex potentials, and $\varepsilon$ is a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator $\mathcal {H}_1=-{d^2}/{\thinspace dx^2 }+V_1(x)$ consists of a pair of isolated eigenvalues and the essential spectrum of the operator $\mathcal {H}_2=-{d^2}/{\thinspace dy^2}+V_2(y)$ has a virtual level at its lower edge and a spectral singularity inside. Additionally, we assume that there is a certain superposition of eigenvalues of the operator $\mathcal {H}_1$ with the virtual level and spectral singularity of the operator $\mathcal {H}_2$; this leads to the emergence of a special threshold in the essential spectrum of the perturbed operator, with the perturbation leading to a bifurcation of this threshold into eigenvalues and resonances with multiplicity doubling. The bifurcation scenario described in this paper is qualitatively different from the previously known ones.