×

Jost functions and Jost solutions for Jacobi matrices. I: A necessary and sufficient condition for Szegő asymptotics. (English) Zbl 1122.47029

Let \(J=J(\{a_n\},\{b_n\})\) be a Jacobi matrix with \(a_n\to 1\), \(b_n\to 0\). The main result of the present paper concerns the Szegő asymptotics for orthonormal polynomials \(p_n\), associated with \(J\). Let \(Q\) be an at most countable set of points \(z\) in the unit disk \(\mathbb{D}\) such that \(z+z^{-1}\) is an eigenvalue of \(J\). Then \(z^n p_n(z+z^{-1})\) converges to a nonzero limit as \(n\to\infty\) uniformly on compact subsets of \(\mathbb{D}\backslash Q\) if and only if the \(a\)’s and \(b\)’s obey the following conditions \[ \sum_{n=1}^\infty | a_n-1| ^2+| b_n| ^2<\infty, \] and both limits \[ \lim_{n\to\infty} \prod_{j=1}^n a_j, \qquad \lim_{n\to\infty} \sum_{j=1}^n b_j \] exist, and the first one is a positive number. The equivalent conditions are given in terms of the spectral data of \(J\). The authors show that the Szegő asymptotics holds at \(z_0\in\mathbb{D}\backslash Q\) if and only if the Jost asymptotics does at the same point.
[Part II has appeared in Int. Math. Res. Not. 2006, No. 5, Art. ID 19396 (2006; Zbl 1122.47030), see the following review.]

MSC:

47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
33C47 Other special orthogonal polynomials and functions
39A70 Difference operators

Citations:

Zbl 1122.47030
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Benzaid, Stud. Appl. Math., 77, 195 (1987) · Zbl 0628.39002
[2] Carleman, Math. Z., 9, 196 (1921) · JFM 48.1249.01
[3] Case, J. Math. Phys., 15, 2166 (1974) · Zbl 0288.42009
[4] Case, J. Math. Phys., 16, 1435 (1975) · Zbl 0304.42015
[5] Christ, Geom. Funct. Anal., 12, 1174 (2002) · Zbl 1039.34076
[6] Coffman, Trans. Am. Math. Soc., 110, 22 (1964)
[7] Damanik, Commun. Math. Phys., 238, 545 (2003) · Zbl 1052.47027
[8] Damanik, Int. Math. Res. Not., 22, 1087 (2004) · Zbl 1084.34075
[9] Damanik, D., Simon, B.: Jost functions and Jost solutions for Jacobi matrices, II. Decay and analyticity. Preprint · Zbl 1122.47030
[10] Deift, Commun. Math. Phys., 203, 341 (1999) · Zbl 0934.34075
[11] Denisov, Geom. Funct. Anal., 14, 529 (2004) · Zbl 1070.47504
[12] Denisov, S., Kupin, S.: Asymptotics of the orthogonal polynomials for the Szegő class with a polynomial weight. To appear in J. Approximation Theory · Zbl 1099.41025
[13] Duren, P.L.: Theory of H^p Spaces. Pure and Applied Mathematics, vol. 38. New York, London: Academic Press 1970 · Zbl 0215.20203
[14] Eastham, M.S.P.: The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem. London Mathematical Society Monographs, New Series, vol. 4. New York: The Clarendon Press, Oxford University Press 1989 · Zbl 0674.34045
[15] Ford, Trans. Am. Math. Soc., 10, 319 (1909) · JFM 40.0384.02
[16] Geronimo, Trans. Am. Math. Soc., 258, 467 (1980)
[17] Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of Mathematical Monographs, vol. 18. Providence, RI: American Mathematical Society 1969 · Zbl 0181.13504
[18] Gončar, Math. USSR Sb., 26, 555 (1975) · Zbl 0341.30029
[19] Harris, J. Math. Anal. Appl., 48, 1 (1974) · Zbl 0304.34043
[20] Hartman, Am. J. Math., 77, 45 (1955) · Zbl 0064.08703
[21] Janas, J. Approximation Theory, 120, 309 (2003) · Zbl 1051.47026
[22] Jost, Helv. Phys. Acta, 20, 256 (1947)
[23] Jost, Phys. Rev., 82, 840 (1951) · Zbl 0042.45206
[24] Katznelson, Y.: An Introduction to Harmonic Analysis. 2nd corrected edn. New York: Dover Publications 1976 · Zbl 0352.43001
[25] Killip, Int. Math. Res. Not., 38, 2029 (2002) · Zbl 1021.34071
[26] Killip, R.; Simon, B., Sum rules for Jacobi matrices and their applications to spectral theory, Ann. Math. (2), 158, 253-321 (2003) · Zbl 1050.47025
[27] Kupin, C. R. Math. Acad. Sci., Paris, 336, 611 (2003) · Zbl 1057.47037
[28] Kupin, Proc. Am. Math. Soc., 132, 1377 (2004) · Zbl 1055.47025
[29] Laptev, Commun. Math. Phys., 241, 91 (2003) · Zbl 1135.47303
[30] Levinson, Duke Math. J., 15, 111 (1948) · Zbl 0040.19402
[31] Nazarov, Int. Math. Res. Not., 3, 155 (2005) · Zbl 1089.47025
[32] Nevai, Mem. Am. Math. Soc., 18, 185 (1979)
[33] Nikishin, J. Sov. Math., 35, 2679 (1986) · Zbl 0605.34023
[34] Peherstorfer, Proc. Am. Math. Soc., 129, 3213 (2001) · Zbl 0976.42012
[35] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness. New York: Academic Press 1975 · Zbl 0308.47002
[36] Rudin, W.: Real and Complex Analysis, 3rd edn. New York: McGraw-Hill 1987 · Zbl 0925.00005
[37] Seiler, Commun. Math. Phys., 42, 163 (1975)
[38] Seiler, J. Math. Phys., 16, 2289 (1975)
[39] Shohat, J.A.: Théorie Générale des Polinomes Orthogonaux de Tchebichef. Mémorial des Sciences Mathématiques, vol. 66, pp. 1-69. Paris 1934
[40] Simon, Adv. Math., 137, 82 (1998) · Zbl 0910.44004
[41] Simon, J. Funct. Anal., 178, 396 (2000) · Zbl 0977.34075
[42] Simon, J. Funct. Anal., 214, 396 (2004) · Zbl 1064.30030
[43] Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory. AMS Colloquium Series. Providence, RI: American Mathematical Society 2005
[44] Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory. AMS Colloquium Series. Providence, RI: American Mathematical Society 2005 · Zbl 1082.42021
[45] Simon, B.: Trace Ideals and Their Applications, 2nd edn. Mathematical Surveys and Monographs. Providence, RI: Amercian Mathematical Society 2005 · Zbl 1074.47001
[46] Simon, Commun. Math. Phys., 242, 393 (2003) · Zbl 1046.42017
[47] Szegő, Math. Ann., 76, 490 (1915) · JFM 45.0518.02
[48] Szegő, Math. Z., 6, 167 (1920) · JFM 48.0376.03
[49] Szegő, Sitzungsber. Berlin Math. Ges., 21, 59 (1922)
[50] Szegő, Math. Ann., 86, 114 (1922) · JFM 48.0378.02
[51] Verblunsky, Proc. Lond. Math. Soc., II. Ser., 40, 290 (1936) · JFM 61.0523.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.