Weyl Theory: Procedures, Stability, Control and Applications

This project is a continuation of the Project P 24301. The following tasks are the principal aims of the project. We plan to develop further those recent spectral and Weyl (Weyl--Titchmarsh) theoretic results on continuous and discrete equations, which either have been obtained in the framework of the Project P 24301 or are related to it. In particular, we plan to find procedures for solving inverse problems for the cases, where uniqueness results were recently obtained, including equations with singularities and various (discrete and continuous) generalizations of Dirac and Schrödinger equations and canonical systems. The most innovative part of the research deals with the study of stability and with regularization for inverse problems of Weyl theory as well as with applications of the methods of Weyl theory to dynamical systems. Some of our initial examples in these directions are presented in the proposal and we assume that these examples admit extensive generalizations and developments. Many inverse problems are unstable and so their regularization is required for applications. Using state space method, method of operator identities and Riccati equations, we will start a study of special cases, where the procedure of solving inverse problems is stable, and we will consider regularization for other cases of inverse problems related to Weyl theory. We will study interconnections between Weyl theory and boundary control method for dynamical Dirac and Schrödinger equations and their generalizations. As a result, we will obtain procedures to recover dynamical systems from response functions for the case of explicit solutions. We will also derive new general type procedures to recover dynamical Dirac and Schrödinger equations from response functions and we will modify these procedures for other important dynamical systems. Using evolution of Weyl functions, we will prolong our study of initial-boundary value problems for integrable wave equations. These problems are mostly overdetermined and we will make emphasis on the study of the cases where boundary conditions are uniquely defined by the initial ones or the other way around. We also expect optimal control results for those cases. We plan to use our version of the generalized Bäcklund-Darboux transformation (GBDT) in order to obtain and study explicit solutions of inverse problems and various explicit (so called multipole) solutions of nonlinear integrable equations. The mentioned above tasks will generate new important results in the related domains of inversion of operators, interpolation, factorization, Riccati equations, spectral theory of functions of (S+N)-triangular operators, and in the, so called, interval stability as well.

Our project is dedicated to Titchmarsh-Weyl (Weyl) theory of classical and nonclassical systems (first aim), stability of solving inverse problems to recover systems from Weyl functions (second aim) and interesting connections with dynamical systems (third aim). We achieved an essential progress in developing further Weyl (spectral) theoretic results. In particular, the A-function theory, taking roots in the brilliant notes of M.G. Krein and developed in the seminal B. Simon and Gesztesy-Simon works for scalar Schrödinger equations, was greatly generalised (jointly with F. Gesztesy) for the case of Dirac systems with rectangular Weyl matrix functions. Moreover, it was shown that for general-type self-adjoint and skew-self-adjoint Dirac systems on the semi-axis, Weyl functions are unique analytic extensions of the reflection coefficients. The case of discrete Dirac systems was also studied. Numerous direct and inverse, spectral and scattering results followed. The cases of singularities (at zero or infinity) were considered. Important results on the fundamental solutions and Weyl functions for canonical systems with 2p 2p Hamiltonians H(x) were derived. Our paper on Toeplitz-block Toeplitz matrices, the structure of their inverses and the recovery of the corresponding reflection coefficients from a certain minimum of information (published in Transactions of AMS) is of interest in signal processing. Inverse problems are, in general, ill-posed. However, we proved that our procedures of explicit solving inverse problems are, with certain modifications, stable in the cases of Dirac systems (self-adjoint and skew-self-adjoint, discrete and continuous). Methods of control and system theories have been actively used. The interconnections between dynamical and spectral Dirac systems, and between the corresponding response and Weyl functions were established. Our GBDT version of Bäcklund-Darboux transformation turned out to be an excellent tool for the construction of explicit solutions of many important dynamical systems. All the aims of the project are fulfilled. Moreover, promising new approaches (methods) were found. Some closely related important and unexpected results were obtained as well. The principle investigator A.L. Sakhnovich published alone and together with co-authors 23 papers in the high quality peer-reviewed international editions. Two more papers appeared in arXiv and are submitted to the journals. The project partly supported three more papers (by Dr. O.R. Popovych). The results have been presented at various seminars and conferences including some invited and plenary talks and an invited lecture series. Some of the results were published in well-known journals on applied mathematics and physics including the results closely related to graphene theory. The project supported active and mutually gainful contacts with our colleagues, cooperation partners and co-authors from Germany, the Netherlands, Japan, UK and USA. The methods and approaches, which appeared during our work on the project, were actively used in our next research proposal.