Boundary Data Maps for Schrodinger Operators on a Compact Interval

TL;DR

This paper systematically analyzes boundary data maps for one-dimensional Schrödinger operators on a compact interval, providing explicit formulas, resolvent relations, and properties in both self-adjoint and non-self-adjoint contexts.

Contribution

It introduces explicit representations and Krein-type formulas for boundary data maps, extending understanding of Schrödinger operators with separated boundary conditions.

Findings

Explicit boundary data map representations in terms of resolvents

Krein-type resolvent formulas for different boundary conditions

Herglotz property of boundary data maps in the self-adjoint case

Abstract

We provide a systematic study of boundary data maps, that is, 2 \times 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrodinger 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 Schrodinger operator and the associated boundary trace maps, Krein-type resolvent formulas relating Schrodinger 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.