Boundary Data Maps for Schrödinger Operators on a Compact Interval

¹ Department of Mathematics & Statistics, Missouri University of Science and Technology Rolla, MO 65409, USA
² Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

^* Corresponding author. E-mail: gesztesyf@missouri.edu

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 Schrödinger 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 Schrödinger operator and the associated boundary trace maps, Krein-type resolvent formulas relating Schrödinger operators corresponding to different (separated) 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.