On Dirichlet-to-neumann Maps and Some Applications to Modified Fredholm Determinants

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrödinger operators in L2(Ω; dnx), where Ω ⊂ Rn, n = 2, 3, 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 modified Fredholm perturbation determinants associated with operators in L2(Ω; dnx) to modified Fredholm perturbation determinants associated with operators in L2(∂Ω; dn−1σ), n = 2, 3. This leads to a twoand three-dimensional 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,∞) to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation.