Opuscula Math. 38, no. 5 (2018), 699-718

Opuscula Mathematica

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

Eduard Ianovich

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 \(L_p[a,b]\) (\(p\ge 1\)). 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 \(g(x)\in L_p[a,b]\) (\(p\ge 1\)) 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 \(f(x)\in L_p[a,b]\).

Keywords: self-adjoint operators, absolutely continuous spectrum, equi-absolute continuity, spectral density, Jacobi matrices.

Mathematics Subject Classification: 47A10, 47A58.

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  • Eduard Ianovich
  • Saint Petersburg, Russia
  • Communicated by Sergey N. Naboko.
  • Received: 2017-11-01.
  • Revised: 2018-01-11.
  • Accepted: 2018-02-21.
  • Published online: 2018-06-12.
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Cite this article as:
Eduard Ianovich, On one condition of absolutely continuous spectrum for self-adjoint operators and its applications, Opuscula Math. 38, no. 5 (2018), 699-718, https://doi.org/10.7494/OpMath.2018.38.5.699

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