On Periodic Matrix-Valued Weyl-Titchmarsh Functions

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 in 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 self-adjoint extension generating 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 Schr\"odinger operator with linear potential and its perturbation by bounded periodic potential

Interpolation theory in Sectorial Stieltjes Classes and Explicit System Solutions

We introduce sectorial classes of matrix-valued Stieltjes functions in which we solve the bitangential interpolation problem of Nudelman and Ball–Gohberg–Rodman. We consider also a new type of solutions of Nevanlinna–Pick interpolation problems, so-called explicit system solutions generated by Brodskii–Livsic colligations, and find conditions on interpolation data of their existence and uniqueness. We point out the connections between sectorial Stieltjes classes and sectorial operators, and find out new properties of the classical Nevanlinna–Pick interpolation matrices (in the scalar case). We present in terms of interpolation data the exact formula for the angle of sectoriality of the main operator in the explicit system solution as well as the criterion for this operator to be extremal.The interpolation model for nonselfadjoint matrices is established

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144