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Boundary Data Maps for Schrödinger Operators on a Compact Interval

Published online by Cambridge University Press:  12 May 2010

S. Clark ,
F. Gesztesy  and
M. Mitrea
S. Clark
Affiliation:
Department of Mathematics & Statistics, Missouri University of Science and Technology Rolla, MO 65409, USA
F. Gesztesy*
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
M. Mitrea*
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
*
* Corresponding author. E-mail: gesztesyf@missouri.edu
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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.

Keywords

(non-self-adjoint) Schrödinger operators on a compact intervalseparated boundary conditionsboundary data mapsRobin-to-Robin mapslinear fractional transformationsKrein-type resolvent formulas
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 73 - 121
Copyright
© EDP Sciences, 2010

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Footnotes

Dedicated with deep admiration to the memory of Mikhail Sh. Birman (1928-2009)

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