Absolutely continuous Jacobi operators

Pedersen, Steen

doi:10.1090/S0002-9939-02-06339-6

Absolutely continuous Jacobi operators
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by Steen Pedersen PDF

Proc. Amer. Math. Soc. 130 (2002), 2369-2376 Request permission

Abstract:

We show (among other results) that a symmetric Jacobi matrix whose diagonal is the zero sequence and whose super-diagonal $h_n>0$ satisfies $h_{2n-1}=h_{2n}$, $h_k\leq h_{k+1}$ and $0<b\leq \tfrac {h_{2k+2}}{k+1}\leq \tfrac {h_{2k}}{k}$ has purely absolutely continuous spectrum when considered as a self-adjoint operator on $\ell ^2(\mathbb {N})$.

References

Ju. M. Berezans′kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. MR 0222718
H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643
S. L. Clark, A spectral analysis for self-adjoint operators generated by a class of second order difference equations, J. Math. Anal. Appl. 197 (1996), no. 1, 267–285. MR 1371289, DOI 10.1006/jmaa.1996.0020
Anne Boutet de Monvel and Jaouad Sahbani, On the spectral properties of the spin-boson Hamiltonians, Lett. Math. Phys. 44 (1998), no. 1, 23–33. MR 1623754, DOI 10.1023/A:1007448732287
Anne Boutet de Monvel and Jaouad Sahbani, On the spectral properties of discrete Schrödinger operators: the multi-dimensional case, Rev. Math. Phys. 11 (1999), no. 9, 1061–1078. MR 1725827, DOI 10.1142/S0129055X99000337
J. Dombrowski and S. Pedersen, Orthogonal polynomials, spectral measures, and absolute continuity, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 115–124. MR 1379123, DOI 10.1016/0377-0427(95)00104-2
J. Dombrowski and S. Pedersen, Spectral measures and Jacobi matrices related to Laguerre-type systems of orthogonal polynomials, Constr. Approx. 13 (1997), no. 3, 421–433. MR 1451714, DOI 10.1007/s003659900051
J. Dombrowski and S. Pedersen, Absolute continuity for Jacobi matrices with constant row sums, J. Math. Anal. Appl., to appear.
Don B. Hinton and Roger T. Lewis, Spectral analysis of second order difference equations, J. Math. Anal. Appl. 63 (1978), no. 2, 421–438. MR 611455, DOI 10.1016/0022-247X(78)90088-4
Svetlana Jitomirskaya and Yoram Last, Power-law subordinacy and singular spectra. I. Half-line operators, Acta Math. 183 (1999), no. 2, 171–189. MR 1738043, DOI 10.1007/BF02392827
J. Janas and M. Moszyński, The alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights, Integral Equations Operator Theory, to appear.
Jan Janas and Serguei Naboko, Jacobi matrices with power-like weights—grouping in blocks approach, J. Funct. Anal. 166 (1999), no. 2, 218–243. MR 1707753, DOI 10.1006/jfan.1999.3434
J. Janas and S. Naboko, Multitreshold spectral phase transition examples in a class of unbounded Jacobi matrices, Recent advances in operator theory (Groningen, 1998), 267–285. Oper. Theory Adv. Appl., 124, Birkhäuser, Basel, 2001.
A. Kiselev and Y. Last, Solutions, spectrum and dynamics for Schrödinger operators on infinite domains, Duke Math. J. 102 (2000), 125–150.
Yoram Last and Barry Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), no. 2, 329–367. MR 1666767, DOI 10.1007/s002220050288
E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys. 78 (1980/81), no. 3, 391–408. MR 603501
C. R. Putnam, Commutation properties of Hilbert space operators and related topics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 36, Springer-Verlag New York, Inc., New York, 1967. MR 0217618
Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR 1711536, DOI 10.1090/surv/072