Spectral properties of the one-dimensional Schrödinger Hamiltonian with non-local $\delta'(x\pm y)$ potentials

Abstract

We consider a model of one-dimensional Schr\"odinger Hamiltonian perturbed by two identical non-local interactions of the form $\delta'(x\pm y)$, symmetrically located at the points $ \pm y$ from the origin. The Schr\"odinger operator under consideration is constructed as a self-adjoint extension of the symmetric Laplacian. The essential spectrum is described, and the condition for the existence of the eigenvalue of the Schr\"odinger operator is investigated. The main results are based on the analysis of the spectral analysis of self-adjoint extension of the operator $\mathbf{h}$.