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Yu. B. Farforovskaya 《Journal of Mathematical Sciences》1987,37(5):1365-1368
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) cf 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. 相似文献
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Given a singular self-adjoint differential operator of order 2n with real coefficients we constructtwo sequences of regular self-adjoint differential expressionsr 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. 相似文献
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L. A. Pastur 《Journal of Mathematical Sciences》1989,46(4):1979-2021
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. 相似文献
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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 相似文献
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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.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1651–1662, December, 2006. 相似文献
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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 maxi|y(xi)-yi|, 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. 相似文献
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Jürgen Voigt 《Journal of Functional Analysis》1977,25(4):317-334
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 cp, 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 cp, 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. 相似文献
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《Journal of Functional Analysis》1987,73(1):122-134
Let Ω denote a connected and open subset of n. The existence of n commuting self-adjoint operators H1,…, Hn on such that each Hj is an extension of (acting on is shown to be equivalent to the existence of a measure μ on n such that f → tf (the Fourier transform of f) is unitary from onto Ω. It is shown that the support of μ can be chosen as a subgroup of n iff H1,…, Hn can be chosen such that the unitary groups generated by H1,…, Hn act multiplicatively on . This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of n by some subgroup, i.e., iff Ω is essentially a fundamental domain. 相似文献
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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. 相似文献
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V. I. Kachalov 《Differential Equations》2013,49(3):386-391
We suggest an algebraic approach to singular perturbation theory and present a generalization of the Poincaré expansion theorem. 相似文献
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Lajos Molnár 《Journal of Mathematical Analysis and Applications》2007,327(1):302-309
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. 相似文献
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L. Tzafriri 《Israel Journal of Mathematics》1966,4(1):62-64
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. 相似文献
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Aiping Wang 《Journal of Mathematical Analysis and Applications》2011,378(2):493-506
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. 相似文献
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V. M. Babich 《Journal of Mathematical Sciences》1997,86(3):2664-2666
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. 相似文献