Jacobi matrices with absolutely

continuous spectrum

Authors:
Jan Janas and Serguei Naboko

Journal:
Proc. Amer. Math. Soc. **127** (1999), 791-800

MSC (1991):
Primary 47B37; Secondary 47B39

MathSciNet review:
1469415

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a Jacobi matrix defined in as , where is a unilateral weighted shift with nonzero weights such that Define the seqences: If and , then has an absolutely continuous spectrum covering . Moreover, the asymptotics of the solution is also given.

**[1]**Ju. M. Berezans′kiĭ,*Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov*, Akademijá Nauk Ukrainskoĭ SSSR. Institut Matematiki, Izdat. “Naukova Dumka”, Kiev, 1965 (Russian). MR**0222719****[2]**H. Behncke,*Absolute continuity of Hamiltonians with von Neumann Wigner potentials. II*, Manuscripta Math.**71**(1991), no. 2, 163–181. MR**1101267**, 10.1007/BF02568400**[3]**Earl A. Coddington and Norman Levinson,*Theory of ordinary differential equations*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR**0069338****[4]**J. Dombrowski,*Cyclic operators, commutators, and absolutely continuous measures*, Proc. Amer. Math. Soc.**100**(1987), no. 3, 457–463. MR**891145**, 10.1090/S0002-9939-1987-0891145-4**[5]**D. J. Gilbert and D. B. Pearson,*On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators*, J. Math. Anal. Appl.**128**(1987), no. 1, 30–56. MR**915965**, 10.1016/0022-247X(87)90212-5**[6]**W. A. Harris Jr. and D. A. Lutz,*Asymptotic integration of adiabatic oscillators*, J. Math. Anal. Appl.**51**(1975), 76–93. MR**0369840****[7]**J. Janas and S.N. Naboko, On the point spectrum of some Jacobi matrices, JOT, to appear.**[8]**S. Khan and D. B. Pearson,*Subordinacy and spectral theory for infinite matrices*, Helv. Phys. Acta**65**(1992), no. 4, 505–527. MR**1179528****[9]**A. Kiselev, Absolute continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly descreasing potentials, Comm. Math. Phys. 179 (1996), 377-400.**[10]**-, Preservation of the absolutely continuous spectrum of Schrödinger equation under perturbations by slowly decreasing potentials and a.e. convergence of integral operators (1997) (preprint).**[11]**Günter Stolz,*Bounded solutions and absolute continuity of Sturm-Liouville operators*, J. Math. Anal. Appl.**169**(1992), no. 1, 210–228. MR**1180682**, 10.1016/0022-247X(92)90112-Q**[12]**Günter Stolz,*Spectral theory for slowly oscillating potentials. I. Jacobi matrices*, Manuscripta Math.**84**(1994), no. 3-4, 245–260. MR**1291120**, 10.1007/BF02567456

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Additional Information

**Jan Janas**

Affiliation:
Institute of Mathematics, Polish Academy of Sciences, Cracow Branch, Sw. Tomasza 30, 31-027 Krakow, Poland

Email:
najanas@cyf-kr.edu.pl

**Serguei Naboko**

Affiliation:
Department of Mathematical Physics, Institute for Physics, St. Petersburg University, Ulianovskaia 1, 198904, St. Petergoff, Russia

Email:
naboko@snoopy.phys.spbu.ru

DOI:
https://doi.org/10.1090/S0002-9939-99-04586-4

Keywords:
Jacobi matrix,
absolutely continuous spectrum,
asymptotics behaviour

Received by editor(s):
June 25, 1997

Additional Notes:
The research of the first author was supported by grant PB 2 PO3A 002 13 of the Komitet Badań Naukowych, Warsaw.

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1999
American Mathematical Society