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Inverse problems for boundary triples with applications

Brown, Brian Malcolm, Marletta, Marco, Wood, I. and Naboko, S. 2017. Inverse problems for boundary triples with applications. Studia Mathematica 237 (3) , pp. 241-275. 10.4064/sm8613-11-2016

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This paper discusses the inverse problem of how much information on an operator can be determined/detected from ‘measurements on the boundary’. Our focus is on non-selfadjoint operators and their detectable subspaces (these determine the part of the operator ‘visible’ from ‘boundary measurements’). We show results in an abstract setting, where we consider the relation between the M- function (the abstract Dirichlet to Neumann map or the transfer matrix in system theory) and the resolvent bordered by projections onto the detectable subspaces. More specifically, we investigate questions of unique determination, reconstruction, analytic continuation and jumps across the essential spectrum. The abstract results are illustrated by examples of Schr¨odinger operators, matrix- differential operators and, mostly, by multiplication operators perturbed by integral oper- ators (the Friedrichs model), where we use a result of Widom to show that the detectable subspace can be characterized in terms of an eigenspace of a Hankel-like operator.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Uncontrolled Keywords: detectable subspace, inverse problem, M-function, Friedrichs model, Widom.
Publisher: Institute of Mathematics of the Polish Academy of Sciences
ISSN: 0039-3223
Funders: Leverhulme Trust RPG167, EU Marie Curie Grant PIIF-GA-2011-299919, Russian Science Foundation Grant 11-15-30007
Date of First Compliant Deposit: 15 February 2017
Date of Acceptance: 27 January 2017
Last Modified: 23 Feb 2020 15:49

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