Self-adjoint Extensions Of Differential Operators Research Articles

Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides

We describe and classify the thresholds of the continuous spectrum and the resulting resonances for general formally self-adjoint elliptic systems of second-order differential equations with Dirichlet or Neumann boundary conditions in domains with cylindrical and periodic outlets to infinity (in waveguides). These resonances arise because there are “almost standing” waves, that is, non-trivial solutions of the homogeneous problem which do not transmit energy. We consider quantum, acoustic, and elastic waveguides as examples. Our main focus is on degenerate thresholds which are characterized by the presence of standing waves with polynomial growth at infinity and produce effects lacking for ordinary thresholds. In particular, we describe the effect of lifting an eigenvalue from the degenerate zero threshold of the spectrum. This effect occurs for elastic waveguides of a vector nature and is absent from the scalar problems for cylindrical acoustic and quantum waveguides. Using the technique of self-adjoint extensions of differential operators in weighted spaces, we interpret the almost standing waves as eigenvectors of certain operators and the threshold as the corresponding eigenvalue. Here the threshold eigenvalues and the corresponding vector-valued functions not decaying at infinity can be obtained by approaching the threshold (the virtual level) either from below or from above. Hence their properties differ essentially from the customary ones. We state some open problems.

Self-adjoint extensions of differential operators on Riemannian manifolds

We study \(H=D^*D+V\), where D is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold M, and V is a Hermitian bundle endomorphism. In the case when M is geodesically complete, we establish the essential self-adjointness of positive integer powers of H. In the case when M is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of H, expressed in terms of the behavior of V relative to the Cauchy boundary of M.

Boundary dynamics and topology change in quantum mechanics

We show how to use boundary conditions to drive the evolution on a quantum mechanical system. We will see how this problem can be expressed in terms of a time-dependent Schrödinger equation. In particular, we will need the theory of self-adjoint extensions of differential operators in manifolds with boundary. An introduction of the latter as well as meaningful examples will be given. It is known that different boundary conditions can be used to describe different topologies of the associated quantum systems. We will use the previous results to study the topology change and to obtain necessary conditions to accomplish it in a dynamical way.

Asymptotics of solutions and modeling of the Von Karman equations in a singularly perturbed domain

In a plane domain with several small holes of diameter O(e), we consider the von Karman equations describing the flexure of a thin isotropic plate. We construct asymptotic expansions for solutions to the nonlinear and the corresponding linearized problems. The coefficients of expansions turn out to be holomorphic and rational functions of |ln e| −1 . The asymptotic results for the linear problem (the Kirchhoff plate) are interpreted within the framework of the theory of selfadjoint extensions of differential operators by using the tool of weighted spaces with separated asymptotics. We also present a model of a nonlinear singularly perturbed problem that provides high accuracy asymptotic formulas. This problem includes the generalized Sobolev conditions at the points to which the holes shrink. Bibliography: 34 titles. Illustrations: 1 figure.

On the regularization of Schrödinger operators with Coulombic singularities

A regularization method will be investigated for the Coulomb problem in three dimensions, which has been mostly applied to classical systems, namely, the Kustaanheimo–Stiefel transform. The regularization is developed within the context of self-adjoint extensions of differential operators and may be used for the semiclassical analysis of low-lying eigenvalues of the Schrödinger operator for one electron in the field of a finite number of fixed nuclei.

Description of self-adjoint extensions of differential operators of arbitrary order on an infinite interval in the absolutely indefinite case

Description of self-adjoint extensions of differential operators of arbitrary order on an infinite interval in the absolutely indefinite case