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

We consider the Schrödinger operator on the plane with bounded potential , where is a real potential, and are compactly supported complex potentials, and is a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator consists of a pair of isolated eigenvalues and the essential spectrum of the operator 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 with the virtual level and spectral singularity of the operator ; 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.