Faddeev differential equations as a spectral problem for a nonsymmetric operator

We consider a nonsymmetric matrix operator whose eigenvalue problem is the system of Faddeev differential equations for a three-particle system. For this operator and its adjoint, the resolvents are represented in terms of Faddeev T-matrix components of the three-particle Schrödinger operator. On the basis of these representations, the invariant spaces of the operators under consideration are investigated and their eigenfunctions are determined. The biorthogonality and completeness of the eigenfunction system are proved.We dedicate this paper to the memory of Stanislav Petrovitch Merkuriev, who left us three years ago.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 513–528, June, 1996.     

On invariant subspaces and eigenfunctions of regular Hecke operators on spaces of multiple theta constants

Invariant subspaces and eigenfunctions of regular Hecke operators acting on spaces spanned by products of even number of Igusa theta constants with rational characteristics are constructed. For some of the eigenfunctions of genuses g=1 and g=2, corresponding zeta functions of Hecke and Andrianov are explicitly calculated.     

The Faddeev equation and the location of the essential spectrum of a model multi-particle operator

In this paper we consider a model operator which acts in a three-particle cut subspace of the Fock space. We describe “two-particle” and “three-particle” branches of the essential spectrum and obtain an analog of the Faddeev equation for eigenfunctions of this operator.     

Spectral properties of Schrödinger operators on radial N-dimensional infinite trees

We study the discreteness of the spectrum of Schrödinger operators which are defined on a class of radial N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their eigenvalues using operators on a one-dimensional tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N-dimensional tree. We show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero. Moreover, the projections to the one-dimensional tree of eigenfunctions of the N-dimensional Laplace operator converge to the corresponding eigenfunctions of the one-dimensional limit operator.     

On infinity of the discrete spectrum of operators in the Friedrichs model

The discrete spectrumof selfadjoint operators in the Friedrichs model is studied. Necessary and sufficient conditions of existence of infinitely many eigenvalues in the Friedrichs model are presented. A discrete spectrum of a model three-particle discrete Schrödinger operator is described.     

Spectral Analysis for Linear Pencils N — λP of Ordinary Differential Operators

Boundary eigenvalue problems for linear pencils N — λ of two ordinary differential operators are studied where P is of lower order than N. In a suitable scale of subspaces of Sobolev spaces and spaces of continuously differentiable functions results on minimality and basis properties of the eigenfunctions and associated functions are proved, including explicit formulas for the Fourier coefficients. As an application the Orr - Sommerfeld equation is considered.     

Operators with smooth functional calculi

We introduce a class of (tuples of commuting) unbounded operators on a Banach space, admitting smooth functional calculi, which contains all operators of Helffer-Sjöstrand type and is closed under the action of smooth proper mappings. Moreover, the class is closed under tensor product of commuting operators. In general, and operator in this class has no resolvent in the usual sense, so the spectrum must be defined in terms of the functional calculus. We also consider invariant subspaces and spectral decompositions.     

The generalized D’Alembert operator on compactified pseudo-Euclidean space

It is proved that the D’Alembert operator in ? ⁿ with multidimensional time, bordered by operators of multiplication by some function, and subject to an acceptance condition at infinity is a self-adjoint operator with discrete spectrum. The spectrum and eigenfunctions are explicitly described.     

Some remarks on analytic continuation in weighted backward shift invariant subspaces

By a famous result of Douglas, Shapiro, and Shields, functions in backward shift invariant subspaces in Hardy spaces are characterized by the fact that they admit a pseudocontinuation outside the closed unit disk. More can be said when the spectrum of the associated inner function has holes on \mathbb T{{\mathbb T}}. Then the functions of the invariant subspaces even extend analytically through these holes. Here we will be interested in weighted backward shift invariant subspaces which appear naturally in the context of kernels of Toeplitz operators. Note that such kernels are special cases of so-called nearly invariant subspaces. In our setting a result by Aleksandrov allows to deduce analytic continuation properties which we will then apply to consider embeddings of weighted invariant subspaces into their unweighted companions. We hope that this connection might shed some new light on known results. We will also establish a link between the spectrum of the inner function and the approximate point spectrum of the backward shift in the weighted situation in the spirit of results by Aleman, Richter, and Ross.     

Dichotomy properties of Hamiltonians for linear systems in extrapolation spaces

For the Hamiltonian operator matrix from systems theory the existence of invariant subspaces corresponding to the spectrum in the right and left half-plane is shown. The control and observation operators are unbounded in the sense that they map into extrapolation spaces, thereby allowing for PDE systems with control and observation on the boundary. The invariant subspaces are then used to construct solutions of the corresponding Riccati equation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Operators associated with soft and hard spectral edges from unitary ensembles

Using Hankel operators and shift-invariant subspaces on Hilbert space, this paper develops the theory of the integrable operators associated with soft and hard edges of eigenvalue distributions of random matrices. Such Tracy-Widom operators are realized as controllability operators for linear systems, and are reproducing kernels for weighted Hardy spaces, known as Sonine spaces. Periodic solutions of Hill's equation give a new family of Tracy-Widom type operators. This paper identifies a pair of unitary groups that satisfy the von Neumann-Weyl anti-commutation relations and leave invariant the subspaces of L² that are the ranges of projections given by the Tracy-Widom operators for the soft edge of the Gaussian unitary ensemble and hard edge of the Jacobi ensemble.     

Perturbation of spectra and spectral subspaces

We consider the problem of variation of spectral subspaces for linear self-adjoint operators with emphasis on the case of off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators, respectively.

     

Normalization of eigenfunctions of the continuous spectrum of a three-dimensional periodic lightguide

Asymptotic formulas for eigenfunctions of the continuous spectrum of a lightguide are given. On solutions of a lightguide, an indefinite inner product is introduced. This inner product is computed for eigenfunctions of the continuous lightguide corresponding to arbitrary, in general, intervals of the continuous spectrum. The result is given in terms of the asymptotic coefficients of the eigenfunctions and the Dirac function. Bibliography: 10 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16, 1997, pp. 68–87.     

The theory of regularized traces of Sturm-Liouville operators as applied to approximate calculation of eigenvalues and eigenfunctions of certain singular operators

The effectiveness of the theory of regularized traces as applied to the approximate calculation of eigenvalues and eigenfunctions is demonstrated for certain singular differential operators. Singular operators of the Bessel type and operators from fluid dynamics and mathematical physics are considered.     

Essential spectrum and exponential estimates of eigenfunctions of lattice Schrödinger and Dirac operators

The main aim of the paper is the investigation of a relation between the essential spectrum and the exponential decay at infinity of eigenfunctions of the lattice analogs of Schrödinger and Dirac operators.     

Hermitian Weighted Composition Operators and Bergman Extremal Functions

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are ${(1 - \overline{w}z)^{-\kappa}}$ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.     

Rodriguez-type Representation for the Eigenfunctions of Durrmeyer-type Operators

In this paper we consider a generalization of different variants of Durrmeyer- type modifications of Baskakov and Meyer- König and Zeller operators. We prove a general result concerning the commutativity of these operators with certain differential operators. Prom this result a Rodriguez- type formula for the eigenfunctions follows as a corollary.     

(NCI)算子与(NFI)算子

引入两类与不变子空间密切相关的算子,即(NCI)算子与(NFI)算子.考虑这两类算子的性质,例如存在性,相似不变性,以及与强不可约算子的关系等.并在可分Banach空间上讨论谱为单点集{0}的算子能否小紧摄动成为(NFI)算子.     

Integrable three-dimensional coupled nonlinear dynamical systems related to centrally extended operator Lie algebras and their Lax type three-linearization

The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Bäcklund transformation. The connection of this hierarchy with integrable by Lax two-dimensional Davey-Stewartson type systems is studied.