Abstract
We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an ℓ p condition and a generalized bounded variation condition. This latter condition requires that a sequence can be expressed as a sum of sequences β (l), each of which has rotated bounded variation, i.e.,
for some \({\phi_l}\) . This includes a large class of discrete Schrödinger operators with almost periodic potentials modulated by ℓ p decay, i.e. linear combinations of \({\lambda_n {\rm cos}(2\pi\alpha n + \phi)}\) with \({\lambda \in \ell^p}\) of bounded variation and any α.
In all cases, we prove absence of singular continuous spectrum, preservation of absolutely continuous spectrum from the corresponding free case, and that pure points embedded in the continuous spectrum can only occur in an explicit finite set.
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Blumenthal, O.: Ueber die Entwicklung einer willkürlichen Funktion nach den Nennern des Kettenbruches für \({\int_{-\infty}^0 \frac{\varphi(\xi) d\xi}{x-\xi}}\) . Ph.D. dissertation, Göttingen, 1898
Breuer, J.: Singular continuous and dense point spectrum for sparse trees with finite dimensions. In: Probability and Mathematical Physics, CRM Proc. and Lecture Notes 42, Providence RI: Amer. Math. Soc., 2007, pp. 65–83
Breuer J.: Spectral and dynamical properties of certain random Jacobi matrices with growing parameter. Trans. Amer. Math. Soc. 362, 3161–3182 (2010)
Breuer J., Last Y., Simon B.: The Nevai condition. Constr. Approx. 32, 221–254 (2010)
Chihara, T. S.: An Introduction to Orthogonal Polynomials. Mathematics and Its Applications 13, New York-London-Paris: Gordon and Breach, 1978
Eggarter T.: Some exact results on electron energy levels in certain one-dimensional random potentials. Phys. Rev. B5, 3863–3865 (1972)
Freud G.: Orthogonal Polynomials. Pergamon Press, Oxford-New York (1971)
Geronimus Ya. L.: Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval. Consultants Bureau, New York (1961)
Geronimus, Ya. L.: Polynomials orthogonal on a circle and their applications. Amer. Math. Soc. Translation 1954(104), 79 pp (1954)
Golinskii L., Nevai P.: Szegő difference equations, transfer matrices and orthogonal polynomials on the unit circle. Commun. Math. Phys. 223, 223–259 (2001)
Gredeskul S. A., Pastur L. A.: Behavior of the density of states in one-dimensional disordered systems near the edges of the spectrum. Theor. Math. Phys. 23, 132–139 (1975)
Janas, J., Simonov, S.: Weyl–Titchmarsh type formula for discrete Schrödinger operator with Wigner–von Neumann potential. To appear in Studia Math. available at http://arxiv.org/abs/1003.3319v1 [math.SP], 2010
Kaluzhny, U., Last, Y.: Purely absolutely continuous spectrum for some random Jacobi matrices. In: Probability and mathematical physics, CRM Proc. Lecture Notes 42, Providence, RI: Amer. Math. Soc., pp. 273–281, 2007
Kaluzhny, U., Shamis, S.: Preservation of Absolutely Continuous Spectrum of Periodic Jacobi Operators Under Perturbations of Square-Summable Variation. to appear in Constr. Approx, available at http://arxiv.org/abs/0912.1142v2 [math.SP], 2010
Kiselev A., Last Y., Simon B.: Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators. Commun. Math. Phys. 194(1), 1–45 (1998)
Last Y.: Destruction of absolutely continuous spectrum by perturbation potentials of bounded variation. Commun. Math. Phys. 274(1), 243–252 (2007)
Máté, A., Nevai, P.: Orthogonal polynomials and absolutely continuous measures. In: Approximation Theory, IV (College Station, TX, 1983), New York: Academic Press, 1983, pp. 611–617
Nevai, P.: Orthogonal polynomials. Mem. Amer. Math. Soc. 18(213), 185 pp, (1979)
Nevai P.: Orthogonal polynomials, measures and recurrences on the unit circle. Trans. Amer. Math. Soc. 300(1), 175–189 (1987)
Nikishin E.M.: An estimate for orthogonal polynomials. Acta Sci. Math. (Szeged) 48(1–4), 395–399 (1985)
Pastur L., Figotin A.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992)
Peherstorfer F., Steinbauer R.: Orthogonal polynomials on the circumference and arcs of the circumference. J. Approx. Theory 102(1), 96–119 (2000)
Prüfer H.: Neue Herleitung der Sturm–Liouvilleschen Reihenentwicklung stetiger Funktionen. Math. Ann. 95(1), 499–518 (1926)
Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory. AMS Colloquium Publications 54.1, Providence, RI: Amer. Math. Soc., 2005
Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory. AMS Colloquium Publications 54.2, Providence, RI: Amer. Math. Soc., 2005
Simon B.: Szegő’s Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials. Princeton University Press, Princeton, NJ (2010)
Simon, B.: Orthogonal polynomials with exponentially decaying recursion coefficients. In: Probability and Mathematical Physics, CRM Proc. Lecture Notes 42 Providence, RI: Amer. Math. Soc., 2007, pp. 453–463
Stieltjes, T.: Recherches sur les fractions continues. Ann. Fac. Sci. Univ. Toulouse 8, J76–J122; ibid. 9, A5–A47 (1894-95)
Szegő, G.: Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ. 23, Providence, RI: Amer. Math. Soc., 1939, third edition, 1967
Verblunsky S.: On positive harmonic functions: A contribution to the algebra of Fourier series. Proc. London Math. Soc. (2) 38, 125–157 (1935)
Weidmann J.: Zur Spektraltheorie von Sturm-Liouville-Operatoren. Math. Z. 98, 268–302 (1967)
Weyl H.: Über beschraänkte quadratische Formen, deren Differenz vollstetig ist. Rend. Circ. Mat. Palermo 27, 373–392 (1909)
Wong M.-W. L.: Generalized bounded variation and inserting point masses. Constr. Approx. 30(1), 1–15 (2009)
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Lukic, M. Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation. Commun. Math. Phys. 306, 485–509 (2011). https://doi.org/10.1007/s00220-011-1287-9
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DOI: https://doi.org/10.1007/s00220-011-1287-9