Magnetic Schrödinger operators with delta‐type potentials

We consider the magnetic anisotropic Schrödinger operator onℝ n1H ϱ , a , W u ( x ) = ( D + a ( x ) ) · ϱ ( x ) ( D + a ( x ) ) + W ( x ) u ( x ) , x ∈ ℝ n , where ϱ ( x ) = ( ϱ i j ( x ) ) i , j = 1 n , D = − i ∇ , a ( x ) = ( a 1 ( x ) , … , a n ( x ) ) is the vector potential of the magnetic field and W ( x ) is the scalar potential of the electric field. We assume that a j and ϱ i j are real‐valued functions belonging to the spaceC b 1 ( ℝ n ) of bounded with first derivatives onℝ nfunctions, whereas W ∈ L ∞ ( ℝ n ) is a complex‐valued electric potential. Let Σ  be a C 2 ‐hypersurface inℝ ndividingℝ non two open domains Ω ± with common boundary Σ.  We assume that Σ  is a closed C 2 ‐hypersurface or unbounded hypersurface of bounded geometry. We consider the magnetic Schrödinger operator   H ϱ , a , W , W s= H ϱ , a , W + W swith singular potentials W s with supports on Σ. We associate withH ϱ , a , W , W san unbounded operator H inL 2 ( ℝ n ) generated by H ϱ , a ,  W with domain in H 2 (Ω + ) ⊕  H 2 (Ω − ) consisting of functions satisfying interaction conditions on Σ. We study the self‐adjointness of the operator H and its Fredholm properties. Moreover, we consider the spectral problem 2H ϱ , a , W + λ W su = 0 , u ∈ H 2 ( Ω + ) ⊕ H 2 ( Ω − ) , λ ∈ ℂ , and reduce this problem to a spectral problem for some boundary pseudodifferential operators on Σ. We describe the domains in ℂ where problem ( 2) has the discrete spectrum.