Third order trace formula

In J. Funct. Anal. 257 (2009) 1092–1132, Dykema and Skripka showed the existence of higher order spectral shift functions when the unperturbed self-adjoint operator is bounded and the perturbation is Hilbert–Schmidt. In this article, we give a different proof for the existence of spectral shift function for the third order when the unperturbed operator is self-adjoint (bounded or unbounded, but bounded below).     

Symmetric anti-eigenvalue and symmetric anti-eigenvector

The idea of symmetric anti-eigenvalue and symmetric anti-eigenvector of a bounded linear operator T on a Hilbert space H is introduced. The structure of symmetric anti-eigenvectors of a self-adjoint and certain classes of normal operators is found in terms of eigenvectors. The Kantorovich inequality for self-adjoint operators and bounds for symmetric anti-eigenvalues for certain classes of normal operators are also discussed.     

Estimation of eigenvalues of self-adjoint operators

Estimates of the number of eigenvalues are obtained for perturbations of certain self-adjoint and unitary operators in a Hilbert space. In particular, we consider a perturbation of the operator of multiplication by an independent variable inL ₂ () andL ₂ (0, 1).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 642–648, May, 1994.     

The projection Kantorovich method for eigenvalue problems

A modified projection method for eigenvalues and eigenvectors of a compact operator T on a Banach space is defined and analyzed. The method is derived from the Kantorovich regularization for second-kind equations involving the operator T. It is shown that when T is a positive self-adjoint operator on a Hilbert space and the projections are orthogonal, the modified method always gives eigenvalue approximations which are at least as accurate as those obtained from the projection method. For self-adjoint operators, the required computation is essentially the same for both methods. Numerical computations for two integral operators are presented. One has T positive self-adjoint, while in the other T is not self-adjoint. In both cases the eigenvalue approximations from the modified method are more accurate than those from the projection method.     

Elementary localization theorems for nonlinear eigenproblems

The minimax formula for linear eigenvalues of a linear operator is used to estimate the parameter values (λ) for which the self-adjoint operator L(λ) on Hilbert space to itself fails to have a bounded inverse. Such λ compose the “nonlinear spectrum” of L. The parameter spaces include regions in real or complex n-space. The localization theorems are used to demonstrate certain necessary conditions for stability of linear integro-partial-differential delay equations.     

Self-adjoint abstract differential operators

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H₁ whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula AE, B(x)0, and the operator B(x) is bounded for all x. An operator L₀ is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) H₁ according to the formula: L₀y=–y + Q(x)y (–0 is a self-adjoint operator in H₁ under the given assumptions.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 65–72, July, 1969.     

Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators

GivenA ₁, the discrete approximation of a linear self-adjoint partial differential operator, the smallest few eigenvalues and eigenvectors ofA ₁ are computed by the homotopy (continuation) method. The idea of the method is very simple. From some initial operatorA ₀ with known eigenvalues and eigenvectors, define the homotopyH(t)=(1–t)A ₀+tA₁, 0t1. If the eigenvectors ofH(t ₀) are known, then they are used to determine the eigenpairs ofH(t ₀+dt) via the Rayleigh quotient iteration, for some value ofdt. This is repeated untilt becomes 1, when the solution to the original problem is found. A fundamental problem is the selection of the step sizedt. A simple criterion to selectdt is given. It is shown that the iterative solver used to find the eigenvector at each step can be stabilized by applying a low-rank perturbation to the relevant matrix. By carrying out a small part of the calculation in higher precision, it is demonstrated that eigenvectors corresponding to clustered eigenvalues can be computed to high accuracy. Some numerical results for the Schrödinger eigenvalue problem are given. This algorithm will also be used to compute the bifurcation point of a parametrized partial differential equation.Dedicated to Herbert Bishop Keller on the occasion of his 70th birthdayThe work of this author was in part supported by RGC Grant DAG93/94.SC30.The work of this author was in part supported by NSF Grant DMS-9403899.     

Expansion of a Self-Adjoint Absolutely Continuous Singular Integral Operator in Generalized Eigenvectors and Its Diagonalization

We describe the relationship between the expansion of a self-adjoint operator in generalized eigenvectors and the direct integral of Hilbert spaces. We perform the explicit diagonalization of a self-adjoint absolutely continuous singular integral operator Y using an Hermitian nonnegative kernel consisting of boundary values of the determining function of the operator T = X + iY with respect to the resolvent of the imaginary part of Y.     

Singular continuous Floquet operator for systems with increasing gaps

Consider the Floquet operator of a time-independent quantum system, periodically perturbed by a rank one kick, acting on a separable Hilbert space: e^−iH₀Te^{−iκT|φ〉〈φ|}, where T and κ are the period and the coupling constant, respectively. Assume the spectrum of the self-adjoint operator H₀ is pure point, simple, bounded from below and the gaps between the eigenvalues (λ_n) grow like λ_n+1−λ_n∼Cn^d with d?2. Under some hypotheses on the arithmetical nature of the eigenvalues and the vector φ, cyclic for H₀, we prove the Floquet operator of the perturbed system has purely singular continuous spectrum.     

Self-adjointness of the dirac operator in a space of vector functions

This paper is devoted to the proof of the self-adjointness of the minimal operator defined on the space L₂(? ∞, ∞; H) (H being a separable Hilbert space) by the expression L=iJ(d/dt)+A+B(t). The coefficients in this expression are self-adjoint operators on H, with A being unbounded, AJ+JA = 0, and the function ∥B(t)∥ H being assumed to lie in L ₂ ^loc (? ∞, ∞). The result obtained is applicable to the Dirac operator.     

A new approach to spectral approximation

A new technique for approximating eigenvalues and eigenvectors of a self-adjoint operator is presented. The method does not incur spectral pollution, uses trial spaces from the form domain, has a self-adjoint algorithm, and exhibits superconvergence.     

Non-self-adjoint Jacobi matrices with a rank-one imaginary part

We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank-one imaginary part. It is shown that given a set of n not necessarily distinct nonreal numbers in the open upper (lower) half-plane uniquely determines an n×n Jacobi matrix with a rank-one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity. Algorithms of reconstruction for such finite Jacobi matrices are presented. A new model complementing the well-known Livsic triangular model for bounded linear operators with a rank-one imaginary part is obtained. It turns out that the model operator is a non-self-adjoint Jacobi matrix. We show that any bounded, prime, non-self-adjoint linear operator with a rank-one imaginary part acting on some finite-dimensional (respectively separable infinite-dimensional Hilbert space) is unitarily equivalent to a finite (respectively semi-infinite) non-self-adjoint Jacobi matrix. This obtained theorem strengthens a classical result of Stone established for self-adjoint operators with simple spectrum. We establish the non-self-adjoint analogs of the Hochstadt and Gesztesy-Simon uniqueness theorems for finite Jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non-self-adjoint tri-diagonal matrices to the case of mixed eigenvalues, real and nonreal. A unique Jacobi matrix, unitarily equivalent to the operator of integration in the Hilbert space L₂[0,l] is found as well as spectral properties of its perturbations and connections with the well-known Bernoulli numbers. We also give the analytic characterization of the Weyl functions of dissipative Jacobi matrices with a rank-one imaginary part.     

Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in <Emphasis Type="Italic">l</Emphasis><Superscript>2</Superscript> by the use of finite submatrices

We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l ²(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J _n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].     

Banach Spaces of Universal Taylor Series in the Disc Algebra

We consider the problem of how to compute eigenvalues of a self-adjoint operator when a direct application of the Galerkin (finite-section) method is unreliable. The last two decades have seen the development of the so-called quadratic methods for addressing this problem. Recently a new perturbation approach has emerged, the idea being to perturb eigenvalues off the real line and, consequently, away from regions where the Galerkin method fails. We propose a simplified perturbation method which requires no á priori information and for which we provide a rigorous convergence analysis. The latter shows that, in general, our approach will significantly outperform the quadratic methods. We also present a new spectral enclosure for operators of the form A + iB where A is self-adjoint, B is self-adjoint and bounded. This enables us to control, very precisely, how eigenvalues are perturbed from the real line. The main results are demonstrated with examples including magnetohydrodynamics, Schrödinger and Dirac operators.     

Discrete Sturm-Liouville Problems With Parameter in the Boundary Conditions

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

Alternative proof of the a priori tan Θ theorem

Let A be a self-adjoint operator in a separable Hilbert space. We assume that the spectrum of A consists of two isolated components σ₀ and σ₁ and the set σ₁ is in a finite gap of the set σ₁. It is known that if V is a bounded additive self-adjoint perturbation of A that is off-diagonal with respect to the partition spec(A) = σ₀ ∪ σ₁, then for \(\left\| V \right\| < \sqrt 2 d\), where d = dist(σ₀, σ₁), the spectrum of the perturbed operator L = A+V consists of two isolated parts ω₀ and ω₁, which appear as perturbations of the respective spectral sets s0 and s1. Furthermore, we have the sharp upper bound ||EA(σ₀) - EL(ω₀)|| ≤ sin (arctan(||V||/d)) on the difference of the spectral projections E_A(σ₀)) and E_L(ω₀)) corresponding to the spectral sets σ₀ and ω₀ of the operators A and L. We give a new proof of this bound in the case where ||V|| < d.     

Properties of the spectrum in the John problem on a freely floating submerged body in a finite basin

We consider a problem on the interaction of surface waves with a freely floating submerged body, which combines a spectral Steklov problem with a system of algebraic equations. We reduce this spectral problem to a quadratic pencil and then to the standard spectral equation for a self-adjoint operator in a certain Hilbert space. In addition to general properties of the spectrum, we investigate the asymptotics of eigenvalues and eigenvectors with respect to an intrinsic small parameter.     

Unimodular eigenvalues and invariant measures for linear operators

For a bounded linear operator in a complex separable Hilbert space we show that it accepts an invariant Borel probability measure of square integrable norm whose essential support spans the space, iff its eigenvectors with unimodular eigenvalues span the space.This research has been supported by the Economic Research Center (KOE) of the Athens University of Economics and Business.