We provide a systematic study of boundary data maps, that is, 2 x 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.
S. L. Clark et al., "Boundary Data Maps for Schrödinger Operators on a Compact Interval," Mathematical Modelling of Natural Phenomena, vol. 5, no. 4, pp. 73-121, EDP Sciences, Jan 2010.
The definitive version is available at https://doi.org/10.1051/mmnp/20105404
Mathematics and Statistics
Keywords and Phrases
Boundary data; Krein-type resolvent formulas; Linear Fractional Transformations; Robin-to-Robin maps; Separated boundary conditions; Linear transformations; Boundary conditions; (Non-self-adjoint) Schrödinger operators on a compact interval; Boundary data maps; Krein-type resolvent formulas; Linear fractional transformations; Separated boundary conditions
International Standard Serial Number (ISSN)
Article - Journal
© 2010 EDP Sciences, All rights reserved.
01 Jan 2010