Alexander Sakhnovich

Weyl–Titchmarsh Theory for Schrödinger Operators with Strongly Singular Potentials

We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly singular potentials such as perturbed spherical Schroedinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a generalized Nevanlinna function. Moreover, we will show how essential supports for the Lebesgue decomposition of the spectral measure can be obtained from the boundary behavior of the singular Weyl function. Finally, we will prove a local Borg-Marchenko type uniqueness result. Our criteria will in particular cover the aforementioned case of perturbed spherical Schroedinger operators.

Inverse eigenvalue problems for perturbed spherical Schrödinger operators

We investigate the eigenvalues of perturbed spherical Schr?dinger operators under the assumption that the perturbation q(x) satisfies xq(x) L1(0, 1). We show that the square roots of eigenvalues are given by the square roots of the unperturbed eigenvalues up to a decaying error depending on the behavior of q(x) near x = 0. Furthermore, we provide sets of spectral data which uniquely determine q(x).

Commutation methods for Schrödinger operators with strongly singular potentials

We explore the connections between singular Weyl–Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the results to spherical Schrodinger operators (also known as Bessel operators). We also investigate the connections with the generalized Backlund–Darboux transformation.

Inverse problems and nonlinear evolution equations : solutions, Darboux matrices and Weyl-Titchmarsh functions

This monograph fits the clearly need for books with a rigorous treatment of the inverse problems for non-classical systems and that of initial-boundary-value problems for integrable nonlinear equations. The authors develop a unified treatment of explicit and global solutions via the transfer matrix function in a form due to Lev A. Sakhnovich. The book primarily addresses specialists in the field. However, it is self-contained and starts with preliminaries and examples, and hence also serves as an introduction for advanced graduate students in the field.

Skew-self-adjoint discrete and continuous Dirac-type systems: inverse problems and Borg–Marchenko theorems

New formulae on the inverse problem for the continuous skew-self-adjoint Dirac-type system are obtained. For the discrete skew-self-adjoint Dirac-type system the solution of a general-type inverse spectral problem is also derived in terms of the Weyl functions. The description of the Weyl functions on the interval is given. Borg–Marchenko-type uniqueness theorems are also derived for both discrete and continuous non-self-adjoint systems.

Semiseparable Integral Operators and Explicit Solution of an Inverse Problem for a Skew-Self-Adjoint Dirac-Type System

Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form

Weyl functions, the inverse problem and special solutions for the system auxiliary to the nonlinear optics equation

{\phi(\lambda)\,{\rm exp}\{-2i{\lambda}D\}}

Recovery of the Dirac system from the rectangular Weyl matrix function

, where