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- Title
ON ONE CONDITION OF ABSOLUTELY CONTINUOUS SPECTRUM FOR SELF-ADJOINT OPERATORS AND ITS APPLICATIONS.
- Authors
Ianovich, Eduard
- Abstract
In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of a self-adjoint operator A by a sequence of operators An with absolutely continuous spectrum on a given interval [a, b] which converges to A in a strong sense on a dense set. The notion of equi-absolute continuity is also used. It was found a sufficient condition of absolute continuity of the operator A spectrum on the finite interval [a, b] and the condition for that the corresponding spectral density belongs to the class Lp[a, b] (p ≥ 1). The application of this method to Jacobi matrices is considered. As one of the results we obtain the following assertion: Under some mild assumptions, suppose that there exist a constant C > 0 and a positive function g(x) ∊ Lp[a, b] (p ≥ 1) such that for all n sufficiently large and almost all x ∊ [a, b] the estimate 1/g(x) ≤ bn(P2n+1(x) + P2n(x)) ≤ C holds, where Pn(x) are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and bn is a second diagonal sequence of Jacobi matrix. Then the spectrum of Jacobi matrix operator is purely absolutely continuous on [a, b] and for the corresponding spectral density f(x) we have f(x) ∊ Lp[a, b].
- Subjects
CONTINUOUS spectrum (Atomic spectrum); ADJOINT operators (Quantum mechanics); APPROXIMATION theory; OPERATOR theory; SET theory; ABSOLUTE continuity
- Publication
Opuscula Mathematica, 2018, Vol 38, Issue 5, p699
- ISSN
1232-9274
- Publication type
Article
- DOI
10.7494/OpMath.2018.38.5.699