On the Negative Eigenvalues of the Discrete Schrödinger Operator with Non-Local Potential in Three-Dimensional Case

Eigenvalue behavior of a family of discrete Schrödinger operators $$H_{\lambda\mu}$$ depending on parameters $$\lambda,\mu\in\mathbb{R}$$ is studied on the three-dimensional lattice $$\mathbb{Z}^{3}$$ . The non-local potential is described by the Kronecker delta function and the shift operator. The existence of eigenvalues below the essential spectrum and their dependence on the parameters are explicitely proven. We also show that the essential spectrum absorbes the threshold eigenvalue and there exists a particular parabola, on whose left intercept the threshold becomes an embedded eigenvalue and the threshold resonance at its other points.

Hali tarjima qilinmagan