On Perturbation of Thresholds in Essential Spectrum under Coexistence of Virtual Level and Spectral Singularity

We study the perturbation of the Schrödinger operator on the plane with a bounded potential of the form $V_1(x)+V_2(y),$ where $V_1$ is a real function and $V_2$ is a compactly supported function. It is assumed that the one-dimensional Schrödinger operator $ \mathcal{H} _1$ with the potential $V_1$ has two real isolated eigenvalues $ \Lambda _0,$$ \Lambda _1$ in the lower part of its spectrum, and the one-dimensional Schrödinger operator $ \mathcal{H} _2$ with the potential $V_2$ has a virtual level at the boundary of its essential spectrum, i.e., at $\lambda=0$, and a spectral singularity at the inner point of the essential spectrum $\lambda=\mu>0$. In addition, the eigenvalues and the spectral singularity overlap in the sense of the equality $ \lambda _0:= \Lambda _0+\mu= \Lambda _1.$ We show that a perturbation by an abstract localized operator leads to a bifurcation of the internal threshold $ \lambda _0$ into four spectral objects which are resonances and/or eigenvalues. These objects correspond to the poles of the local meromorphic continuations of the resolvent. The spectral singularity of the operator $ \mathcal{H} _2$ qualitatively changes the structure of these poles as compared to the previously studied case where no spectral singularity was present. This effect is examined in detail, and the asymptotic behavior of the emerging poles and corresponding spectral objects of the perturbed Schrödinger operator is described. DOI 10.1134/S106192084010059