Inequalities for means of chords, with application to isoperimetric problems

We consider a pair of isoperimetric problems arising in physics. The first concerns a Schr\"odinger operator in $L^2(\mathbb{R}^2)$ with an attractive interaction supported on a closed curve $\Gamma$, formally given by $-\Delta-\alpha \delta(x-\Gamma)$; we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread $\Gamma$ in $\mathbb{R}^3$, homogeneously charged but not conducting, and we ask about the (renormalized) potential-energy minimizer. Both problems reduce to purely geometric questions about inequalities for mean values of chords of $\Gamma$. We prove an isoperimetric theorem for $p$-means of chords of curves when $p \leq 2$, which implies in particular that the global extrema for the physical problems are always attained when $\Gamma$ is a circle. The article finishes with a discussion of the $p$--means of chords when $p > 2$.Comment: LaTeX2e, 11 page

The absolutely continuous spectrum of one-dimensional Schr"odinger operators

This paper deals with general structural properties of one-dimensional Schr"odinger operators with some absolutely continuous spectrum. The basic result says that the omega limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas.Comment: references added; a few very minor change

Asymptotic behaviour of the spectrum of a waveguide with distant perturbations

We consider the waveguide modelled by a $n$-dimensional infinite tube. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators ''localized'' in a certain sense, and the distance between their ''supports'' tends to infinity. We study the asymptotic behaviour of the discrete spectrum of such system. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We also provide some particular examples of the distant perturbations. The examples are the potential, second order differential operator, magnetic Schroedinger operator, curved and deformed waveguide, delta interaction, and integral operator

A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators

Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered.Comment: 38 pages, Proposition 2.2 and its proof corrected, Remarks 2.5, 3.4, and 3.12 extended, details added in subsections 2.3 and 4.2, section 6 rearranged, typos corrected, references adde

Schrödinger operators with δ and δ′-potentials supported on hypersurfaces

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity

Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms

We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples

Around the Van Daele–Schmüdgen Theorem

For a {bounded} non-negative self-adjoint operator acting in a complex, infinite-dimensional, separable Hilbert space H and possessing a dense range R we propose a new approach to characterisation of phenomenon concerning the existence of subspaces M\subset H such that M\capR=M^\perp\capR=\{0\}. We show how the existence of such subspaces leads to various {pathological} properties of {unbounded} self-adjoint operators related to von Neumann theorems \cite{Neumann}--\cite{Neumann2}. We revise the von Neumann-Van Daele-Schm\"udgen assertions \cite{Neumann}, \cite{Daele}, \cite{schmud} to refine them. We also develop {a new systematic approach, which allows to construct for any {unbounded} densely defined symmetric/self-adjoint operator T infinitely many pairs of its closed densely defined restrictions T_k\subset T such that \dom(T^* T_{k})=\{0\} (\Rightarrow \dom T_{k}^2=\{0\}$) k=1,2 and \dom T_1\cap\dom T_2=\{0\}, \dom T_1\dot+\dom T_2=\dom T

New aspects of Krein`s extension theory

The extension problem for closed symmetric operators with a gap is studied. A new kind of parametrization of extensions (the so-called Krein model) is developed. The notion of a singular operator plays the key role in our approach. We give the explicit description of extensions and establish the spectral properties of extended operators.Досліджується проблема розширень замкнених симетричних операторів із щілиною в спектрі. Розвинуто новий спосіб параметризації розширень (так звана модель Крейна). Ключову роль у нашому підході відіграє поняття сингулярного оператора. Дано явний опис розширень і встановлені спектральні властивості розширених операторів

Boundary Triplets, Tensor Products and Point Contacts to Reservoirs

We consider symmetric operators of the form S: = A⊗ IT+ IH⊗ T, where A is symmetric and T= T∗ is (in general) unbounded. Such operators naturally arise in problems of simulating point contacts to reservoirs. We construct a boundary triplet Π S for S∗ preserving the tensor structure. The corresponding γ-field and Weyl function are expressed by means of the γ-field and Weyl function corresponding to the boundary triplet Π A for A∗ and the spectral measure of T. An application to 1-D Schrödinger operators is given. A model of electron transport through a quantum dot assisted by cavity photons is proposed. In this model the boundary operator is chosen to be the well-known Jaynes–Cummings operator which is regarded as the Hamiltonian of the quantum dot. © 2018, Springer International Publishing AG, part of Springer Nature