Inverse problems for boundary triples with applications

B. M. Brown¹ ; M. Marletta² ; S. Naboko³ ; I. Wood⁴

¹ Cardiff School of Computer Science and Informatics Cardiff University Queen’s Buildings 5 The Parade Cardiff CF24 3AA, UK
² School of Mathematics Cardiff University Senghennydd Road Cardiff CF24 4AG, UK
³ Department of Mathematical Physics Institute of Physics St. Petersburg State University 1 Ulianovskaia, St. Petergoff St. Petersburg, 198504, Russia
⁴ School of Mathematics, Statistics and Actuarial Science University of Kent Cornwallis Building Canterbury CT2 7NF, UK

Studia Mathematica, Tome 237 (2017) no. 3, pp. 241-275

Résumé

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 Schrödinger 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.

Détail

DOI : 10.4064/sm8613-11-2016

Keywords: paper discusses inverse problem much information operator determined detected measurements boundary focus non selfadjoint operators their detectable subspaces these determine part operator visible boundary measurements results abstract setting where consider relation between m function abstract dirichlet neumann map transfer matrix system theory resolvent bordered projections detectable subspaces specifically investigate questions unique determination reconstruction analytic continuation jumps across essential spectrum abstract results illustrated examples schr dinger operators matrix differential operators mostly multiplication operators perturbed integral operators friedrichs model where result widom detectable subspace characterized terms eigenspace hankel like operator