We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the A−amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl m−function associated with the Schrödinger operator H = −∂2 x + q (x) on L2 (0,∞) with Dirichlet boundary condition at x = 0.

This article is concerned with the Titchmarsh–Weyl m ( ) function for the di erential equation dy=dx+[ −q(x)]y=0. The test potential q(x) = x, for which the relevant m ( ) functions are meromorphic, having simple poles at the points = 4k + 1 and = 4k + 3, is studied in detail. We are able to calculate the m ( ) function both far from and near to these poles. The calculation is then extended to ...

We show that the multitude of applications of the Weyl–Titchmarsh m-function leads to a multitude of di4erent functions in the theory of orthogonal polynomials on the unit circle that serve as analogs of the m-function. c © 2004 Elsevier B.V. All rights reserved.

We consider 2×2–Hamiltonian systems of the form y′(t) = zJH(t)y(t), t ∈ [s−, s+). If a system of this form is in the limit point case, an analytic function is associated with it, namely its Titchmarsh–Weyl coefficient qH . The (global) uniqueness theorem due to L. de Branges says that the Hamiltonian H is (up to reparameterization) uniquely determined by the function qH . In the present paper w...

We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl -Titchmarsh matrixvalued functions as well as a new version of the functional model in such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-v...

In this paper we present an overview of the basic Weyl–Titchmarsh theory for second order Sturm–Liouville equations on time scales. We construct the m(λ)function, the Weyl solution, and the Weyl disk. We justify the terminology “disk” by its geometric properties, show explicitly the coordinates of the center of the disk, and calculate its radius. We show that the dichotomy regarding the squarei...

We investigate the singular Weyl–Titchmarsh m-function of perturbed spherical Schrödinger operators (also known as Bessel operators) under the assumption that the perturbation q(x) satisfies xq(x) ∈ L(0, 1). We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function bel...