# Inverse problems for boundary triples with applications

Brown, Brian Malcolm, Marletta, Marco, Naboko, Serguei, Wood, Ian (2017) Inverse problems for boundary triples with applications. Studia Mathematica, 237 . pp. 241-275. ISSN 0039-3223. E-ISSN 1730-6337. (doi:10.4064/sm8613-11-2016) (KAR id:60783)

## Abstract

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 operators(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 10.4064/sm8613-11-2016 detectable subspace, inverse problem, M-function, Friedrichs model, Widom Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, CalculusQ Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science Ian Wood 09 Mar 2017 15:30 UTC 16 Feb 2021 13:43 UTC https://kar.kent.ac.uk/id/eprint/60783 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-7181-7075