Boundary Data Maps for Schrödinger Operators on a Compact Interval

S. Clark; F. Gesztesy; M. Mitrea

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 4, page 73-121
  • ISSN: 0973-5348

Abstract

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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.

How to cite

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Clark, S., Gesztesy, F., and Mitrea, M.. "Boundary Data Maps for Schrödinger Operators on a Compact Interval." Mathematical Modelling of Natural Phenomena 5.4 (2010): 73-121. <http://eudml.org/doc/197614>.

@article{Clark2010,
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.},
author = {Clark, S., Gesztesy, F., Mitrea, M.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {(non-self-adjoint) Schrödinger operators on a compact interval; separated boundary conditions; boundary data maps; Robin-to-Robin maps; linear fractional transformations; Krein-type resolvent formulas; Kreĭn-type resolvent formulas},
language = {eng},
month = {5},
number = {4},
pages = {73-121},
publisher = {EDP Sciences},
title = {Boundary Data Maps for Schrödinger Operators on a Compact Interval},
url = {http://eudml.org/doc/197614},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Clark, S.
AU - Gesztesy, F.
AU - Mitrea, M.
TI - Boundary Data Maps for Schrödinger Operators on a Compact Interval
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/5//
PB - EDP Sciences
VL - 5
IS - 4
SP - 73
EP - 121
AB - 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.
LA - eng
KW - (non-self-adjoint) Schrödinger operators on a compact interval; separated boundary conditions; boundary data maps; Robin-to-Robin maps; linear fractional transformations; Krein-type resolvent formulas; Kreĭn-type resolvent formulas
UR - http://eudml.org/doc/197614
ER -

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