Boundary Data Maps for Schrödinger Operators on a Compact Interval

Abstract

We provide a systematic study of boundary data maps, that is, 2 £ 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schr¨ odinger operators on a compact interval (0;R) with separated boundary conditions at 0 and R. Most of our results are formulated in the non-self-adjoint context. Our principal results include explicit representations of these boundary data maps in terms of the resolvent of the underlying Schr¨ odinger operator and the associated boundary trace maps, Krein-type resolvent formulas relating Schr¨ odinger operators corresponding to different (sepa- rated) boundary conditions, and a derivation of the Herglotz property of boundary data maps (up to right multiplication by an appropriate diagonal matrix) in the special self-adjoint case.