Inverse problems for boundary triples with applications
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.