Magnetic Schrödinger operators with delta‐type potentials

Abstract

We consider the magnetic anisotropic Schrödinger operator on where is the vector potential of the magnetic field and W(x) is the scalar potential of the electric field. We assume that aj and ϱij are real‐valued functions belonging to the space of bounded with first derivatives on functions, whereas is a complex‐valued electric potential. Let Σ be a C2‐hypersurface in dividing on two open domains Ω± with common boundary Σ. We assume that Σ is a closed C2‐hypersurface or unbounded hypersurface of bounded geometry. We consider the magnetic Schrödinger operator with singular potentials Ws with supports on Σ. We associate with an unbounded operator in generated by Hϱ,a, W with domain in H2(Ω+) ⊕ H2(Ω−) consisting of functions satisfying interaction conditions on Σ. We study the self‐adjointness of the operator and its Fredholm properties. Moreover, we consider the spectral problem and reduce this problem to a spectral problem for some boundary pseudodifferential operators on Σ. We describe the domains in where problem (2) has the discrete spectrum.