Two-dimensional Schrödinger operators with non-local singular potentials

In this paper we introduce and study a family of self-adjoint realizations of the Laplacian in $L^2\left(R^2\right)$ with a new type of transmission conditions along a closed bi-Lipschitz curve Σ. These conditions incorporate jumps in the Dirichlet traces both of the functions in the operator domains and of their Wirtinger derivatives and are non-local. Constructing a convenient generalized boundary triple, they may be parametrized by all compact self-adjoint operators in $L^2(\Sigma ;C^2)$. Whereas for all choices of parameters the essential spectrum is stable and equal to $[0,+\infty )$, the discrete spectrum exhibits diverse behavior. While in many cases it is finite, we will describe also a class of parameters for which the discrete spectrum is infinite and accumulates at −∞. The latter class contains a non-local version of the oblique transmission conditions. Finally, we will connect the current model to its relativistic counterpart studied recently in [33].