Math. Model. Nat. Phenom.
Volume 5, Number 4, 2010Spectral problems. Issue dedicated to the memory of M. Birman
|Page(s)||73 - 121|
|Published online||12 May 2010|
Boundary Data Maps for Schrödinger Operators on a Compact Interval
Department of Mathematics & Statistics, Missouri University of
Science and Technology Rolla, MO
2 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
* Corresponding author. E-mail:
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.
Mathematics Subject Classification: 34E05 / 34B20 / 34L40 / 34A55
Key words: (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
© EDP Sciences, 2010
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