Lipschitz functions of self-adjoint operators in perturbation theory

Let A be a self-adjoint operator in a Hubert space. In order that for each differentiable function f and for each self-adjoint operator B one should have the estimate f(B)–f(A) c_f B–A it is necessary and sufficient that the spectrum. of the operator A be a finite set. If m is the number of points of the spectrum of the operator A, then for the constant cf one can take 8(log₂m+2)² [f], where [f] is the Lipschitz constant of the function f.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 176–182, 1985.     

Regular approximation of singular self-adjoint differential operators

Given a singular self-adjoint differential operator of order 2n with real coefficients we constructtwo sequences of regular self-adjoint differential expressions^r which converge to ina generalized sense of resolvent convergence. The first constructionis suitable when no information about the real resolvent setof is available. The second is suitablewhen we know a real point of the resolvent set of .The main application of this construction is in numerical solutionof singular differential equations.     

Spectral theory of random self-adjoint operators

The survey reviews recent results on spectral analysis of differential and finite-difference operators with random spatially homogeneous coefficients. The corresponding problems that crystallized in the development of a number of areas in mathematics and related sciences are very rich and diverse. We discuss the traditional problems of spectral analysis, where the use of probabilistic ideas and methods now allows highly detailed spectral analysis to be performed for an essentially broader class of operators, as well as new problems and results obtained in the framework of this theory.Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 25, pp. 3–67, 1987.     

Initial-value technique for self-adjoint singular perturbation boundary value problems

We have developed an initial-value technique for self-adjoint singularly perturbed two-point boundary value problems. The original problem is reduced to its normal form, and the reduced problem is converted into first-order initial-value problems. These initial-value problems are solved by the cubic spline method. Numerical illustrations are given at the end to demonstrate the efficiency of our method. Graphs are also depicted in support of the results. An erratum to this article can be found at     

Jacobi matrices associated with the inverse eigenvalue problem in the theory of singular perturbations of self-adjoint operators

We establish the relationship between the inverse eigenvalue problem and Jacobi matrices within the framework of the theory of singular perturbations of unbounded self-adjoint operators. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1651–1662, December, 2006.     

Geometric mesh FDM for self-adjoint singular perturbation boundary value problems

A numerical method based on finite difference method with variable mesh is given for second order singularly perturbed self-adjoint two point boundary value problems. The original problem is reduced to its normal form and the reduced problem is solved by FDM taking variable mesh(geometric mesh). The maximum absolute errors max_i|y(x_i)-y_i|, for different values of parameter , number of points N, and the mesh ratio r, for three examples have been given in tables to support the efficiency of the method.     

Perturbation theory for commutative m-tuples of self-adjoint operators

Generalizing the Weyl-von Neumann theorem for normal operators, we show that a commutative m-tuple of self-adjoint operators in a separable Hilbert space may be changed into a diagonal one by adding compact perturbations of class c_p, for p>m. On the other hand it is shown that the absolutely continuous part, defined appropriately, of a commutative m-tuple of self-adjoint operators is stable under perturbations of class c_p, if p < m, m ? 3, or if p = 1, m = 2 (the latter case m = 2 corresponding to the case of one normal operator). For the proof of these Kato-Rosenblum-type theorems a wave operator method for m-tuples is introduced.     

Spectral theory of commuting self-adjoint partial differential operators

Let Ω denote a connected and open subset of $\text{R}$ⁿ. The existence of n commuting self-adjoint operators H₁,…, H_n on $\text{L}{}^2\text{(Ω)}$ such that each H_j is an extension of $\text{−}\text{i∂}\text{∂x}{}_{\mathrm{j}}$ (acting on $\text{C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(Ω))}$ is shown to be equivalent to the existence of a measure μ on $\text{R}$ⁿ such that f → $\text{}\text{̂}$tf (the Fourier transform of f) is unitary from $\text{L}{}^2\text{(Ω)}$ onto Ω. It is shown that the support of μ can be chosen as a subgroup of $\text{R}$ⁿ iff H₁,…, H_n can be chosen such that the unitary groups generated by H₁,…, H_n act multiplicatively on $\text{L}{}^2\text{(Ω)}$. This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of $\text{R}$ⁿ by some subgroup, i.e., iff Ω is essentially a fundamental domain.     

Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators H₀, H and discuss the spectral projections of these operators corresponding to the interval (−∞,λ). The purpose of the paper is to study the spectral properties of the difference D(λ) of these spectral projections. We completely describe the absolutely continuous spectrum of the operator D(λ) in terms of the eigenvalues of the scattering matrix S(λ) for the operators H₀ and H. We also prove that the singular continuous spectrum of the operator D(λ) is empty and that its eigenvalues may accumulate only at “thresholds” in the absolutely continuous spectrum.     

On the algebraic fundamentals of singular perturbation theory

We suggest an algebraic approach to singular perturbation theory and present a generalization of the Poincaré expansion theorem.     

Elementary operators on self-adjoint operators

Let H be a Hilbert space and let A and B be standard ∗-operator algebras on H. Denote by As and Bs the set of all self-adjoint operators in A and B, respectively. Assume that and are surjective maps such that M(AM^∗(B)A)=M(A)BM(A) and M^∗(BM(A)B)=M^∗(B)AM^∗(B) for every pair A∈As, B∈Bs. Then there exist an invertible bounded linear or conjugate-linear operator and a constant c∈{−1,1} such that M(A)=cTAT^∗, A∈As, and M^∗(B)=cT^∗BT, B∈Bs.     

On perturbation theory for spectral operators

Conditions are given under which a Rellich’s perturbation theorem for normal operators on Hilbert spaces may be generalized for spectral operators on Banach spaces.     

The classification of self-adjoint boundary conditions of differential operators with two singular endpoints

For general even order linear ordinary differential equations with real coefficients and endpoints which are regular or singular and for arbitrary deficiency index d, the self-adjoint domains are determined by d linearly independent boundary conditions. These conditions are of three types: separated, coupled, and mixed. We give a construction for all conditions of each type and determine the number of conditions of each type possible for a given self-adjoint domain. Our construction gives a direct alternative to the recent construction of Everitt and Markus which uses the theory of symplectic spaces. We believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self-adjoint boundary conditions. Such forms are known only in the second order, i.e. Sturm-Liouville, case. Even for regular problems of order four no such forms are available. In the case when all d conditions are separated this construction yields explicit non-real conditions for all orders greater than two. It is well known that no such conditions exist in the second order case.     

Natural oscillations of mechanical systems and the spectral theory of self-adjoint operators

In this paper, the following assertion is illustrated with some examples. In practically all cases, the problem on natural oscillations for linear holonomic nondissipative mechanical systems reduces to the problem of determining eigenvalues and eigenelements of the corresponding self-adjoint operator acting in the Hilbert space generated by the quadratic form of the kinetic energy. Bibliography: 1 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 12–16. This work was supported by the Russian Foundation of Fundamental Research (Grant 93-011-16148). Translated by N. S. Zabanikova.