On periodic matrix-valued Weyl–Titchmarsh functions

Abstract

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 matrix-valued functions as well as a new version of the functional model for such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions, we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its selfadjoint extension which generate a periodic Weyl-Titchmarsh matrix-valued function. We study pairs of operators (a symmetric operator and its self-adjoint extension) with constant Weyl-Titchmarsh matrix-valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in Weyl's form. As a consequence of such an approach, we obtain the Stone-von Neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups. Our examples include multiplication operators in weighted spaces, first and second order differential operators, as well as the Schrödinger operator with linear potential and its perturbation by bounded periodic potential.  2004 Elsevier Inc. All rights reserved.

Automorphic Invariant Isometric Operators

“…, one obtains θ(ḡ(z)) = θ(z), that is the function θ defined by (6.3) is automorphic with respect to the group generated by transformationḡ 0 . In other form this result was obtained in [8]. and…”

“…Let H be a closed densely defined symmetric operator on a Hilbert space H with finite defect numbers. Assume that the operator H is (s, U )-periodic (see [8] [3]), that is…”

“…(This example was considered in [8].) The Cayley transform of H is the isometry V with domain D defined as…”

On Self-Adjoint Extensions and Symmetries in Quantum Mechanics

“…In a different vein we quote the spectral analysis of Hamiltonians in concentric spherical shells where the spherical symmetry is used in a critical way [18,17]. In an operator theoretic context we refer, for example, to the notion of periodic Weyl-Titchmarsh functions or invariant operators with respect to linear-fractional transformations [8,7]. Even from a purely geometric viewpoint we should mention the analysis of isospectral manifolds in the presence of symmetries [37].…”