Finite Rank Perturbations and Model Theory

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Many physical systems are modeled by differential operators. One way of describing a system's long-term behavior is through spectral theory, which includes the finding of frequencies naturally exhibited by the system. Imagine a vibrating string or beam of fixed length. Clearly, its frequency (think "sound") depends on how its ends (aka boundaries) are clamped down or otherwise restricted. In general, the spectrum of a differential operator and with it the properties of a physical system can change drastically when the conditions imposed on the boundary are changed. In many cases, we know the complete spectrum for one set of boundary conditions. From there, we can gain knowledge about the system under other conditions via the theory of so-called finite rank perturbations. The primary goal of the project is to systematically study spectral theory of finite rank unitary perturbations by tightening the relationship to corresponding functional models. The rank one setting is reasonably well-understood, while a general treatment of the finite rank problem presented several digressions. Initially, non-cyclic unitary unperturbed operators posed an issue. This problem is now resolved. The general problem involves matrix-valued Herglotz and Cauchy-type transforms. The study of these transforms as part of this project will provide insight into finite rank perturbation theory. A regularization of the so-called exterior Cauchy transform (as was done in the rank one setting) is planned. Further investigations will be made into the difficult case of infinite rank perturbations with defect operators in the trace class. The analogous self-adjoint setting, as well as concrete applications to differential operators, will also be investigated.