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

https://doi.org/10.7494/OpMath.2018.38.5.699

Opuscula Mathematica

# 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 \(L_p[a,b]\) (\(p\ge 1\)). 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.

- N.I. Akhiejzer, The Classical Moment Problem, New-York, Hafner, 1965.
- N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Space, Dover Publications, Inc. New York, 1993.
- A. Aptekarev, J. Geronimo, Measures for orthogonal polynomials with unbounded recurrence coefficients, arXiv:1408.5349.
- A. Aptekarev, J. Geronimo, Measures for orthogonal polynomials with unbounded recurrence coefficients, J. Approx. Theory 207 (2016), 339-347.
- S. Clark, A spectral analysis for self-adjoint operators generated by a class of second order difference equations, J. Math. Anal. Appl. 197 (1996) 1, 267-285.
- J. Dombrowski, Cyclic operators, commutators, and absolutely continuous measures, Proc. Amer. Math. Soc. 100 (1987) 3, 457-463.
- J. Geronimo, W. Van Assche, Approximating the weight function for orthogonal polynomials on several intervals, J. Approx. Theory 65 (1991) 3, 341-371.
- E.A. Ianovich, Jacobi matrices: continued fractions, approximation, spectrum, arXiv:1707.04695.
- J. Janas, M. Moszynski, Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights, Integral Equations Operator Theory 43 (2002), 397-416.
- J. Janas, S. Naboko, Jacobi matrices with power-like weights-grouping in blocks approach, J. Funct. Anal. 166 (1999), 218-243.
- J. Janas, S. Naboko, Asymptotics of generalized eigenvectors for unbounded Jacobi matrices with power-like weights, Pauli matrices commutation relations and Cesaro averaging, Oper. Theory Adv. Appl. 117 (2000), 165-186.
- J. Janas, S. Naboko, Multithreshold spectral phase transitions for a class of Jacobi matrices, Oper. Theory Adv. Appl. 124 (2001), 267-285.
- A. Mate, P. Nevai, Orthogonal polynomials and absolutely continuous measures, [in:] C.K. Chui et al., Approximation IV, Academic Press, New York, 1983, pp. 611-617.
- M. Moszynski, Spectral properties of some Jacobi matrices with double weights, J. Math. Anal. Appl. 280 (2003), 400-412.
- I.P. Natanson, Theory of Functions of a Real Variable, vol. 1, Frederick Ungar Publishing Company, New York, 1964.
- I.I. Privalov, Boundary Properties of Analytic Functions, Gostehizdat, Moscow, 1950.
- V.I. Smirnov, Course of Higher Mathematics, vol. 5, Gostehizdat, Moscow, 1959.
- G. Stolz, Spectral theory for slowly oscillating potentials, I. Jacobi matrices, Manuscripta Math. 84 (1994), 245-260.
- G. Świderski, Periodic perturbations of unbounded Jacobi matrices II: Formulas for density, J. Approx. Theory 216 (2017), 67-85.
- G. Szego, Orthogonal Polynomials, American Mathematical Society, Providence, Rhode Island, 1939.
- R. Szwarc, Absolute continuity of certain unbounded Jacobi matrices. Advanced Problems in Constructive Approximation, [in:] M.D. Buhmann, D.H. Mache (eds.), International Series of Numerical Mathematics, vol. 142, 2003, pp. 255-262.
- P. Turán, On the zeros of the polynomials of Legendre, Časopis Pěst Mat. Fys. 75 (1950), 113-122.

- 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.