This proposal consists of several parts. Part one concerns Aleksandrov-Clark Theory, which relates rank one perturbations to functional models, the Hilbert transform, holomorphic composition operators, rigid functions and the Nehari interpolation problem. A formula (a generalization of the normalized Cauchy transform) for the adjoint of the Clark operator has recently been proven by the principal investigator in collaboration with Salem Prize winner S. Treil. It should be possible to gain some control over the embedded singular spectrum for rank one perturbations, a long standing problem in the field. Part two pertains to the interpretation of a wide class of singular integral operators, including those of Calderon-Zygmund type, in the two-weight situation with very general measures (in particular, non-doubling). Singular integral operators play an essential role in modern Analysis. The last two parts are devoted to aspects of cyclicity. Cyclicity is related to the question for which measures the corresponding Hardy space is dense in that of square integrable functions with respect to the measure, to the backward shift operator, and a classical result of Douglas, Shapiro and Shields connects cyclicity with pseudocontinuation; and thus opening a new area of complex function theory. In part three, the goal is to prove that (assuming cyclicity of the operator) any non-zero vector yields cyclic vectors for rank one perturbations for almost all parameters. This may be useful in practice. It should be mentioned, that for a cyclic operator, it may not be easy to find a cyclic vector. Two interesting new notions are introduced for the setting of an operator on a separable Hilbert space: Subcyclic vectors (a refinement of the definition of cyclic vectors) and a certain graph. Apart from studying their properties, a deep relationship between them in the context of Anderson-type Hamiltonians has preliminarily been proven by the principal investigator in collaboration with E. Abakumov and A. Poltoratski. The last part of this project is connected to the famous problem of Anderson localization, which was suggested by Nobel laureate P. W. Anderson in 1958. The PI will study the cyclicity of vectors for Anderson-type Hamiltonians (a generalization of most Anderson models, e.g. random Schroedinger operators) via analytical as well as numerical methods.
The underlying goal is to develop the mathematical tools necessary to understand the dynamics of physical systems. Such systems are often described by second-order differential equations, like the Schroedinger equation from quantum mechanics (the governing mechanics at the molecular level) and the string equation which is the cutting edge attempt to unite quantum mechanics with general relativity (the theory describing gravity). One of the objects of study - 'singular integral operators' - have become a useful tool in perturbation theory. The latter is concerned roughly with the following question: Given certain information about a physical system, can one predict what happens in the case where one parameter, for example in the initial condition, is changed/perturbed? Cyclicity, for physicists, means that the spectrum (e.g. of light) is simple or non-degenerate. In many problems it is important to know whether this is the case or not. For example, the above-mentioned Anderson localization addresses the question whether or not an impure crystal allows the diffusion of waves or, roughly speaking, whether all electrons stay within a bounded region in space. The results obtained in the scope of this project will be published in scientific journals and reported at research conferences. The proposed subjects provide a wealth of accessible research questions for undergraduate and graduate students. The principal investigator will write expository articles, give seminars at the student level, and mentor young researchers interested in the many open problems she will make available.
