Eigenvalues of Schrödinger Operators with Complex Surface Potentials

We consider Schrödinger operators in R with complex potentials supported on a hyperplane and show that all eigenvalues lie in a disk in the complex plane with radius bounded in terms of the L norm of the potential with d − 1 < p ≤ d. We also prove bounds on sums of powers of eigenvalues. Introduction and main results. Recently there has been great interest in bounds on eigenvalues of Schrödinger operators with complex potentials. A conjecture of Laptev and Safronov [19] states that for a certain range of p’s, all eigenvalues of a Schrödinger operator lie in a disk in the complex plane whose radius is bounded from above in terms of only the L norm of the potential. This conjecture was motivated by a corresponding result by Abramov, Aslanyan and Davies [1] in one dimension and with p = 1. In one part of the parameter regime the conjecture was proved in [9], and in the other part it was proved in [15] for radial potentials. For arbitrary potentials it is still open. In this paper we deal with the analogue of this question for potentials supported on a hyperplane, which is a special case of what is called a ‘leaky graph Hamiltonian’ in [8]. More specifically, in R, d ≥ 2, we introduce coordinates x = (x, xd) with x ∈ R and xd ∈ R and consider the Schrödinger operator −∆+ σ(x′)δ(xd) in L(R) (1) with a complex function σ on R. If σ ∈ L(R) for some p > 1 in d = 2 and p ≥ d − 1 in d ≥ 3, this formal expression can be given meaning as an m-sectorial operator in L(R) through the quadratic form