Extensions Of Symmetric Operators Research Articles

Point interaction of two distinguishable particles in an external field

We consider a point interaction model for two particles in an external field, which is similar to the Skornyakov-Ter-Martirosyan model and is treated using the theory of self-adjoint extensions of symmetric operators. The corresponding energy operator has an infinite set of bound states with the energy values receding to-∞.

On a hyperstokeslet

Correct mathematical description of zero-range interaction for poliharmonic operator is given both for the case of singularity inside a domain and at the boundary. The approach is based on the theory of self-adjoint extensions of symmetric operators.

Solvable model of a double quantum electron layer in a magnetic field

A solvable model of a double quantum electron layer in a magnetic field based on the theory of self-adjoint extensions of symmetric operators is suggested. The spectral problem reduces to a finite-...

Spectral analysis of dissipative Schrödinger operators

Dissipative Schrodinger operators are studied in L2(0, ∞) which are extensions of symmetric operators with defect index (2, 2). We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix according to the scheme of Lax and Phillips. With the help of the incoming spectral representation, we construct a functional model of the dissipative operator and construct its characteristic function in terms of solutions of the corresponding differential equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operator.

Lax–Phillips theory for scattering by thin elastic bodies

The scheme of construction of linear self-adjoint operators for acoustic systems with an interaction of sound and vibration is proposed. Kinetic energy of a system induces the basic Hilbert space L2(Ω), where Ω denotes the domain filled with fluid. Potential energy of the system induces the quadratic form on this Hilbert space and, consequently, the positive operator that governs the evolution. Unlike earlier theories, our operators are perturbations of the Laplace operator in L2 spaces. The operators may also be obtained as extensions of symmetric operator (−Δ) defined on C∞0(Ω) into a wider Hilbert space. Functions from their domains have an additional smoothness of their derivatives on the boundary; the smoothness follows from the elasticity of the boundary. The scheme allows one to describe inhomogeneities of vibrating objects concentrated at a point or on a line (such as cracks and stiffeners). It is demonstrated that Lax–Phillips scattering theory may be generalized for scattering by thin elastic objects. The necessary condition for the existence of the incoming and the outgoing subspaces is derived in terms of spectral properties of the operator of the isolated elastic object.

Indefinite metric and scattering by a domain with a small hole

For the problem of plane waves scattered by a domain with a small hole, we suggest a model based on the theory of self-adjoint extensions of symmetric operators in a space with indefinite metric. For two-dimensional problems of scattering on a line with a hole and on a semi-ellipse connected by a hole with a half-plane, we justify the choice of extension that guarantees the coincidence of the model solution with the solution of the “actual” problem in the far zone with a high degree of accuracy.

Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

This is a brief survey of M. G. Krein's contribution to the theory of self-adjoint extensions of Hermitian operators and to the theory of boundary-value problems for differential equations. The further development of these results is also considered.

Multiparameter spectral problems and commuting extensions of symmetric operators

Multiparameter spectral problems and commuting extensions of symmetric operators

Algebraic version of extension theory for a quantum system with internal structure

An explicit two-channel matrix representation is obtained for the Hamiltonians of systems with internal structure in the framework of the theory of extensions of symmetric operators. Generalized potentials that are equivalent to zero-range interactions with internal structure and automatically reproduce the boundary conditions that correspond to them asx→0 are constructed.

Interpolation problems, extensions of symmetric operators and reproducing kernel spaces II

The aim of part I and this paper is to study interpolation problems for pairs of matrix functions of the extended Nevanlinna class using two different approaches and to make explicit the various links between them. In part I we considered the approach via the Krein-Langer theory of extensions of symmetric operators. In this paper we adapt Dym's method to solve interpolation problems by means of the de Branges theory of Hilbert spaces of analytic functions. We also show here how the two solution methods are connected.

Interpolation problems, extensions of symmetric operators and reproducing kernel spaces II

Interpolation problems, extensions of symmetric operators and reproducing kernel spaces II

Method of boundary integral equations for the second boundary-value problem in domains with a cut

The solution of the second boundary value problem in R3 with a cut reduces to an integrodifferential equation, for the solving of which one can use operator methods of the theory of extensions of symmetric operators. The bounded solvability of the corresponding operator equation problem is proved.

Theory of the self-adjoint extensions of symmetric operators with a spectral gap

Theory of the self-adjoint extensions of symmetric operators with a spectral gap

Extensions of symmetric operators with distinct deficiency numbers

Extensions of symmetric operators with distinct deficiency numbers

Self-adjoint extensions of symmetric operators

Let ℋ denote the Hilbert space of analytic functions on the unit disk which are square summable with respect to the usual area measure. In this paper we consider the formal differential exepressons of order two or greater having the form {fx321-1} and {fx321-2} which give rise to symmetric operators in ℋ. We show that these operators in ℋ admit self-adjoint extensions in ℋ.

Nth order Stieltjes differential boundary operators and Stieltjes differential boundary systems

This article discusses linear differential boundary systems, which include nth-order differential boundary relations as a special case, in L n p[0,1] × L n p[0,1], 1 ⩽ p < ∞. The adjoint relation in L n q[0,1] × L n q[0,1], 1 p + 1 q = 1 , is derived. Green's formula is also found. Self-adjoint relations are found in L n 2[0,1] × L n 2[0,1], and their connection with Coddington's extensions of symmetric operators on subspaces of L n p[0,1] × L n 2[0,1] is established.