This paper has two main goals. First, we are concerned with a description of all self-adjoint extensions of the Laplacian $$ - \Delta {|_{C_0^\infty (\Omega )}}$$
 in L
 2(Ω; d
 n
 x). Here, the domain Ω belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), that contains all convex domains as well as all domains of class C
 1,r
 , for r > 1/2. Second, we establish Kreĭn-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasiconvex domains and study the well-posedness of boundary value problems for the Laplacian as well as basic properties of the corresponding Weyl-Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps). One significant innovation in this paper is an extension of the classical boundary trace theory for functions in spaces that lack Sobolev regularity in a traditional sense, but are suitably adapted to the Laplacian.

A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains

/pdf/boundary-value-problems-weyl-functions-and-differential-40c8fa5ghk.pdf

Boundary Value Problems, Weyl Functions, and Differential Operators

We study generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schr\"odinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$. We also discuss the case of bounded $C^{1,r}$-domains, $(1/2)<r<1$.

Generalized Robin Boundary Conditions, Robin-to-Dirichlet Maps, and Krein-Type Resolvent Formulas for Schrödinger Operators on Bounded Lipschitz Domains

We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[ 
\tau f = \frac{1}{r} (- \big(p[f' + s f]\big)' + s p[f' + s f] + qf),] where the coefficients $p$, $q$, $r$, $s$ are real-valued and Lebesgue measurable on $(a,b)$, with $p\neq 0$, $r>0$ a.e.\ on $(a,b)$, and $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$, and $f$ is supposed to satisfy [f \in AC_{\text{loc}}((a,b)), \; p[f' + s f] \in AC_{\text{loc}}((a,b)).] In particular, this setup implies that $\tau$ permits a distributional potential coefficient, including potentials in $H^{-1}_{\text{loc}}((a,b))$. 
We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator $T_{\text{max}}$, or equivalently, all self-adjoint extensions of the minimal operator $T_{\text{min}}$, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of $T_{\text{min}}$. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira $m$-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of $T_{\text{min}}$. 
Finally, in the special case where $\tau$ is regular, we characterize the Krein-von Neumann extension of $T_{\text{min}}$ and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).

Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

We develop a new approach and present an independent solution to von Neumann’s problem on the parametrization in explicit form of all nonnegative self-adjoint extensions of a densely defined nonnegative symmetric operator. Our formulas are based on the Friedrichs extension and also provide a description for closed sesquilinear forms associated with nonnegative self-adjoint extensions. All basic results of the well-known Krein and Birman-Vishik theory and its complementations are derived as consequences from our new formulas, including the parametrization (in the framework of von Neumann’s classical formulas) for all canonical resolvents of nonnegative selfadjoint extensions. As an application all nonnegative quantum Hamiltonians corresponding to point-interactions in $\mathbb{R}^3$ are described.

The von Neumann Problem for Nonnegative Symmetric Operators

The solution of von Neumann's problem about parametrization of all nonegative selfadjoint extensions of a nonnegative densely defined operator in terms of his formulas is obtained.

/pdf/on-von-neumann-s-problem-in-extension-theory-of-nonnegative-1bgrt06vl6.pdf

On von Neumann’s problem in extension theory of nonnegative operators

Passive systems $\tau = \{T,{\mathfrak{M}},{\mathfrak{N}},{\mathfrak{H}}\}$ with ${\mathfrak{M}}$ and ${\mathfrak{N}}$ as an input and output space and ${\mathfrak{H}}$ as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system $\tau$ with ${\mathfrak{M}} = {\mathfrak{N}}$ is said to be quasi-selfadjoint if ran $(T - T^*) \subset {\mathfrak{N}}$. The subclass ${\bf S}^{qs}({\mathfrak{N}})$ of the Schur class ${\bf S}({\mathfrak{N}})$ is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass ${\bf S}^{qs}({\mathfrak{N}})$ is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass ${\bf S}^{qs}({\mathfrak{N}})$ and the Q-function of T is given.

Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

Passive linear systems τ = 
$${\{A, B, C, D; \mathfrak{H}, \mathfrak{M}, \mathfrak{N}\}}$$

 have their transfer function 
$${\Theta_{\tau}(\lambda) = D + \lambda C{(I - \lambda A)^{-1}}B}$$

 in the Schur class S
 
$${(\mathfrak{M}, \mathfrak{N})}$$

 . Using a parametrization of contractive block operators the transfer function 
$${\Theta_{\tau}(\lambda)}$$

 is connected to the Sz.-Nagy–Foias characteristic function 
$${\Phi_{A}(\lambda)}$$

 of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions 
$${\Theta_{\tau}(\lambda)}$$

 and 
$${\Phi_{A}(\lambda)}$$

 . The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained.

/pdf/parametrization-of-contractive-block-operator-matrices-and-4strmbwhig.pdf

Parametrization of Contractive Block Operator Matrices and Passive Discrete-Time Systems

In this paper operator-valued Q-functions of Krein-Ovcharenko type are introduced. Such functions arise from the extension theory of Hermitian contractive operators A in a Hilbert space ℌ. The definition is related to the investigations of M.G. Krein and I.E. Ovcharenko of the so-called Qμ- and QM-functions. It turns out that their characterizations of such functions hold true only in the matrix valued case. The present paper extends the corresponding properties for wider classes of selfadjoint contractive extensions of A. For this purpose some peculiar but fundamental properties on the behaviour of operator ranges of positive operators will be used. Also proper characterizations for Qμ- and QM-functions in the general operator-valued case are given. Shorted operators and parallel sums of positive operators will be needed to give a geometric understanding of the function-theoretic properties of the corresponding Q-functions.

/pdf/q-functions-of-hermitian-contractions-of-krein-ovcharenko-5gio1u9k9y.pdf

Yury Arlinskii

Papers

The von Neumann Problem for Nonnegative Symmetric Operators

On von Neumann’s problem in extension theory of nonnegative operators

Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

Parametrization of Contractive Block Operator Matrices and Passive Discrete-Time Systems

Q-functions of Hermitian contractions of Krein-Ovcharenko type