arXiv subjects
Publications and source records attributed to Eduard Ianovich.
In this paper it is considered a spectral density for a class of Jacobi matrices with absolutely continuous spectrum that was examined first by Moszynski. It is shown that the corresponding spectral density is equivalent to the positive continuous function everywhere except maybe the point $x=0$.
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 $f(x)\in L_p[a,b\,]$.
In this work the spectral theory of self-adjoint operator $A$ represented by Jacobi matrix is considered. The approach is based on the continued fraction representation of the resolvent matrix element of $A$. Different criteria of absolute continuity of a spectrum are found. For the analysis of the absolutely continuous spectrum it is used an approximation of $A$ 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. In the case when $[a,b\,]\subseteqσ(A)$ it was found the sufficient condition of absolute continuity of the operator $A$ spectrum on $[a,b\,]$. This condition uses the notion of equi-absolute continuity. It is constructed the system of functions converging to the distribution function of the operator. In the case of the absolutely continuous spectrum, the system of continuous functions converging to the spectral weight of the operator on a given interval is also constructed and was analyzed. The conditions when the derivative of the distribution function of $A$ belongs to the class $C[a,b\,]$ are also obtained.