Inverse problems for Hankel operators

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The proposed research belongs to the area of Spectral Theory. Generally speaking, the aim of Spectral Theory is to study in rigorous mathematical terms the connection between structural and geometric properties of rigid bodies on the one hand, and frequencies and patterns of vibrations (oscillations) of these bodies on the other hand. Oscillations in question may be mechanical, acoustic, elastic, electromagnetic, or may involve microscopic particles. Mathematically, this usually reduces to the study of the discrete spectrum (i.e. eigenvalues) of linear operators. The mathematical study of Spectral Theory usually reduces to the analysis of very general properties that are common to a large number of systems. In particular, a crucial role is played by model systems; these are idealized mathematical objects designed to emulate some important features of the real physical systems, yet being sufficiently simple so that they are amenable to rigorous mathematical analysis. The proposed research focuses on one such model system: Hankel operators. Hankel operators are idealized mathematical models that reveal deep structural links between Spectral Theory and other areas of mathematics: Function Theory and Complex Analysis. Broadly speaking, the problems in Spectral Theory can be classified into direct and inverse problems. Direct problem: "Given a linear operator of a certain class, how to determine its spectrum?" Inverse problem: "Given a the spectrum of a linear operator of a certain class, how to reconstruct the operator?" Inverse problems have been popularised by M.Kac through his famous question "Can one hear the shape of a drum?"The aim of the project is to study the inverse problem for Hankel operators. More precisely: every Hankel operator has a set of eigenvalues; these eigenvalues (together with some additional parameters) are called the spectral data. We aim to study the one-to-one correspondence between the Hankel operators and their spectral data.