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- W4297904103 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 self-adjoint operator $A$ by a sequence of operators $A_n$ 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 $L_p[a,b,]$ ($pge 1$). The application of this method to Jacobi matrices is considered. As a one of the results we obtain the following assertion: Under some mild assumptions (see details in Theorem (2.4)), suppose that there exist a constant $C>0$ and a positive function $g(x)in L_p[a,b,]$ ($pge1$) such that for all $n$ sufficiently large and almost all $xin[a,b,]$ the estimate $frac{displaystyle 1}{displaystyle g(x)}le b_n(P_{n+1}^2(x)+P_{n}^2(x))le C$ holds, where $P_n(x)$ are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and $b_n$ 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 $f(x)in L_p[a,b,]$." @default.
- W4297904103 created "2022-10-01" @default.
- W4297904103 creator A5038591236 @default.
- W4297904103 date "2017-11-02" @default.
- W4297904103 modified "2023-10-14" @default.
- W4297904103 title "On one condition of absolutely continuous spectrum for self-adjoint operators and its applications" @default.
- W4297904103 doi "https://doi.org/10.48550/arxiv.1711.00621" @default.
- W4297904103 hasPublicationYear "2017" @default.
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