Hardy spaces on multiply-connected domains and their applications to spectral theory

The spectrum of an operator on Hilbert space is a closed subset of the complex plane.  On the spectral complement, there is a family of analytic functions, called resolvent functions, associated to the operator.  A natural inverse question follows: given a closed subset of the plane and a resolvent-type function on its complement, to what extent can we recover the associated operator? This question leads naturally to the study of Hardy spaces on multiply-connected domains.  In this talk, we survey some results applying this perspective to Schroedinger and related operators.