On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

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 $L_p[a,b\,]$ ($p\ge 1$). 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 $g(x)\in L_p[a,b\,]$ ($p\ge1$) such that for all $n$ sufficiently large and almost all $x\in[a,b\,]$ 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 $f(x)\in L_p[a,b\,]$.