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 a 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 $L_{p}[a,b]$ ($p\geq 1$). 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 $C\gt 0$ and a positive function $g(x)\in L_p[a,b]$ ($p\geq 1$) such that for all n sufficiently large and almost all $x\in [a,b]$ the estimate $\frac{1}{g(x)}\le b_n(P_{n+1}^2(x)+P_{n}^2(x))\le C$ 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 $f(x)\in L_p[a,b]$.