Weyl Theory and Initial-Boundary Value Problems

We shall develop further the classical Weyl theory for canonical systems, Krein systems, and some other self- adjoint systems. Moreover, we shall develop Weyl theory for equations with singularities and for non-self-adjoint equations, like radial Dirac equations, perturbed spherical Schrödinger equations, various classes of equations rationally depending on the spectral parameter, discrete systems, and block Jacobi matrices. We shall develop and apply the approach to inverse problems, which goes back to M.G. Krein and is based on the inversion of the bounded and positive operators, which have some simple structure and satisfy operator identities. In the case of the self-adjoint Dirac-type systems this was treated by M.G. Krein, the corresponding operators were operators with difference kernels, which were recovered from spectral functions. For the non-self-adjoint cases, where generalized Weyl matrix functions do not belong to the Herglotz class, we shall recover the corresponding structured operators directly from the Weyl functions. Related topics of the triangular operator factorization and linear similarity of operators will be treated too. Using a special version of the Bäcklund-Darboux transformation we are going to treat explicitly inverse problems for the case of rational Weyl matrix functions (and some more general classes of Weyl functions) as well. Explicit construction of the fundamental solutions is closely related to the explicit recovery of the potentials from the Weyl functions, that is, to the explicit solution of the inverse problems. Such fundamental solutions will be constructed. We shall apply our special version of the Bäcklund-Darboux transformation to various discrete and continuous integrable equations. Using (additionally) the so called S-multinodes we shall study equations with $k>1$ space variables. We expect various applications to analysis, random matrix theory, electromagnetics, and quantum mechanics. To study the initial-boundary value problems we shall use an Inverse Spectral Transform approach, where Weyl theory is of basic importance. We plan to develop this approach much further. In particular, we are going to develop further the applications of the Inverse Spectral Transform to the second harmonic generation model and to apply Weyl theory to the classical Korteweg-de Vries equation, generalizations of nonlinear Schrödinger equation, Camassa-Holm equation, Hashimoto flow on a discrete ark, and Heisenberg magnet chain. Finally, we plan to study the long-time asymptotics of the solutions via a Riemann-Hilbert approach.

Weyl theory is an important and actively developing part of the spectral theory of differential equations and systems of differential equations. Direct problems of Weyl theory deal with the Weyl functions (or matrix functions) whereas inverse problems are connected with the recovery of differential equations and systems of differential equations from the Weyl functions. In the framework of the project we developed further Weyl theory for classical Dirac and Schrodinger equations and introduced Weyl theory for important non-classical equations. In particular, we solved the inverse problem (formulated by F. Gesztesy) to recover the classical Dirac system with locally square summable potentials from the Weyl functions. The result is valid for the nonclassical case of rectangular Weyl matrix functions as well. Moreover, in a joint work with J. Eckhardt, F. Gesztesy, R. Nichols and G. Teschl, we applied this result for solving inverse problem for Schrödinger-type operators with distributional matrix-valued potentials.In the framework of the project, we developed also Weyl theory for the non-classical discrete self-adjoint and skew-self-adjoint Dirac systems including the case of rectangular Weyl matrix functions.We use two different approaches: one for solving general-type inverse problems and another for explicit solving of inverse problems in the case of rational Weyl matrix functions. The stability of explicit solving inverse problems, which is important in applications, was proved for discrete and continuous Dirac systems.Using our results on evolution of Weyl functions we studied initial-boundary value problems for integrable nonlinear wave equations. Since these initial-boundary value problems are often overdetermined, it is very important to reduce the necessary initial- boundary data. Under some simple conditions we showed how to recover boundary data from the initial data (and vice versa) for the well-known nonlinear Schrödinger equation. This approach admits generalization for other integrable wave equations.In addition, the developed methods were applied to inverse problems for dynamical Dirac system, to the construction of explicit solutions of non-stationary Dirac and Schrodinger equations, of various nonlinear wave equations and of dynamical systems, and to the study of stable solutions of the spatially inhomogeneous case of nonlinear Fokker-Planck equation.The results of the project are described in a monograph, one review paper and thirteen research papers.