On Dirichlet-to-Neumann Maps, Nonlocal Interactions, and Some Applications to Fredholm Determinants

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrödinger operators describing nonlocal interactions in L^2 (Ω; d^(n)x), where Ω ⊂ R^n, n ∈ N, n ≥ 2,are open sets with a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of Fredholm perturbation determinants associated with operators in L^2 (Ω; d^(n)x) to Fredholm perturbation determinants associated with operators in L^2(∂Ω; d^(n-1)σ), n ∈ N, n ≥ 2 . This leads to an extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schrödinger operator on the half-line (0, ∞), in the case of local interactions, to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation.