Fine structure of the zeros of orthogonal polynomials IV: A priori bounds and clock behavior

Communications on Pure and Applied Mathematics

Volumn 61, Issue 4, 2008, Pages 486-538

  Last, Yoram ^a   Simon, Barry ^b  

^a HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 52349091524     PISSN: 00103640     EISSN: 00103640     Source Type: Journal    
DOI: 10.1002/cpa.20185     Document Type: Article

References (55)

1. Avron, J.; Simon, B. Almost periodic Schrödinger operators. II. The integrated density of states. Duke Math. J. 50 (1983), no. 1, 369-391.
2. Avron, J.; van Mouche, P. M. H.; Simon, B. On the measure of the spectrum for the almost Mathieu operator. Comm. Math. Phys. 132 (1990), no. 1, 103-118.
3. Cantero, M. J.; Moral, L.; Velázquez, L. Measures and para-orthogonal polynomials on the unit circle. East J. Approx. 8 (2002), no. 4, 447-464.
4. Cantero, M. J.; Moral, L.; Velázquez, L. Measures on the unit circle and unitary truncations of unitary operators. J. Approx. Theory 139 (2006), no. 1-2, 430-468.
5. Carmona, R. Exponential localization in one-dimensional disordered systems. Duke Math. J. 49 (1982), no. 1, 191-213.
6. Damanik, D.; Simon, B. Jost functions and Jost solutions for Jacobi matrices. II. Decay and analyticity. Int. Math. Res. Not. 2006, Art. ID 19396, 32 pp.
7. Davies, E. B.; Simon, B. Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle. J. Approx. Theory 141 (2006), 189-213.
8. Deift, P. A.; Simon, B. Almost periodic Schrödinger operators. III. The absolutely continuous spectrum in one dimension. Comm. Math. Phys. 90 (1983), no. 3, 389-411.
9. Erdös, P.; Turán, P. On interpolation. III. Interpolatory theory of polynomials. Ann. of Math. (2) 41 (1940), 510-553.
10. Freud, G. Orthogonal polynomials. Pergamon, Oxford-New York, 1971.
11. Geronimus, L. Ya. Orthogonal polynomials: Estimates, asymptotic formulas, and series of polynomials orthogonal on the unit uircle and on an interval. Consultants Bureau, New York, 1961.
12. Gilbert, D. J. On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints. Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), no. 3-4, 213-229.
13. Gilbert, D. J.; Pearson, D. B. On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators. J. Math. Anal. Appl. 128 (1987), no. 1, 30-56.
14. Golinskii, L. Quadrature formula and zeros of para-orthogonal polynomials on the unit circle. Acta Math. Hungar. 96 (2002), no. 3, 169-186.
15. Hairer, E.; Wanner, G. Analysis by its history. Undergraduate Texts in Mathematics. Readings in Mathematics. Springer, New York, 1996.
16. Harrell, E.; Simon, B. The mathematical theory of resonances whose widths are exponentially small. Duke Math. J. 47 (1980), no. 4, 845-902.
17. Hundertmark, D.; Simon, B. Lieb-Thirring inequalities for Jacobi matrices. J. Approx. Theory 118 (2002), no. 1, 106-130.
18. Jitomirskaya, S.; Last, Y. Dimensional Hausdorff properties of singular continuous spectra. Phys. Rev. Lett. 76 (1996), no. 11, 1765-1769.
19. Jitomirskaya, S.; Last, Y. Power-law subordinacy and singular spectra. I. Half-line operators. Acta Math. 183 (1999), no. 2, 171-189.
20. Jitomirskaya, S.; Last, Y. Power law subordinacy and singular spectra. II. Line operators. Comm. Math. Phys. 211 (2000), no. 3, 643-658.
21. Jitomirskaya, S.; Schulz-Baldes, H.; Stolz, G. Delocalization in random polymer models. Comm. Math. Phys. 233 (2003), no. 1, 27-48.
22. Johnson, R.; Moser, J. The rotation number for almost periodic potentials. Comm. Math. Phys. 84 (1982), no. 3, 403-438.
23. Kahn, S.; Pearson, D. B. Subordinacy and spectral theory for infinite matrices. Helv. Phys. Acta 65 (1992), no. 4, 505-527.
24. Killip, R.; Kiselev, A.; Last, Y. Dynamical upper bounds on wavepacket spreading. Amer. J. Math. 125 (2003), no. 5, 1165-1198.
25. Killip, R.; Simon, B. Sum rules for Jacobi matrices and their applications to spectral theory. Ann. of Math. (2) 158 (2003), 253-321.
26. 2 potentials. Ann. of Math., in press.
27. Killip, R.; Stoiciu, M. Eigenvalue statistics for CMV matrices: From Poisson to clock via circular beta ensembles. Preprint, 2006. arXiv: math-ph/0608002, 2006.
28. Kirsch, W.; Martinelli, F. On the density of states of Schrödinger operators with a random potential. J. Phys. A 15 (1982), no. 7, 2139-2156.
29. Kooman, R. J. Asymptotic behaviour of solutions of linear recurrences and sequences of Mobius-transformations. J. Approx. Theory 93 (1998), no. 1, 1-58.
30. Last, Y. On the measure of gaps and spectra for discrete ID Schrödinger operators. Comm. Math. Phys. 149 (1992), no. 2, 347-360.
31. Minami, N. Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Comm. Math. Phys. 177 (1996), no. 3, 709-725.
32. Molčanov, S. A. The local structure of the spectrum of the one-dimensional Schrödinger operator. Comm. Math. Phys. 78 (1980/81), no. 3, 429-446.
33. Nevai, P. G. Orthogonal polynomials. Mem. Amer. Math. Soc. 18 (1979), no. 213.
34. Nevai, P. Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48 (1986), no. 1, 167 pp.
35. Newton, R. G. Scattering theory of waves and particles. 2nd ed. Springer, New York, 1982.
36. Pan, K. Asymptotics for orthogonal polynomials beyond the analytic boundary. Rocky Mountain J. Math. 26 (1996), no. 1, 269-279.
37. Pastur, L. A. Spectra of random selfadjoint operators. (Russian) Uspehi Mat. Nauk 28 (1973), no. 1(169), 3-64.
38. Simon, B. Fine structure of the zeros of orthogonal polynomials. II. OPUC with competing exponential decay. J. Approx. Theory 135 (2005), no. 1, 125-139.
39. Simon, B. Fine structure of the zeros of orthogonal polynomials. III. Periodic recursion coefficients. Comm. Pure Appl. Math. 59 (2005), no. 7, 1042-1062.
40. Simon, B. Meromorphic Szego functions and asymptotic series for Verblunsky coefficients. Acta Math. 195 (2005), 267-285.
41. Simon, B. Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications, 54. American Mathematical Society, Providence, R.I., 2005.
42. Simon, B. Orthogonal polynomials on the unit circle. Part 2. Spectral theory. American Mathematical Society Colloquium Publications, 54. American Mathematical Society, Providence, R.I., 2005.
43. Simon, B. Fine structure of the zeros of orthogonal polynomials. I. A tale of two pictures. Electron. Trans. Numer. Anal. 25 (2006), 328-368.
44. Simon, B. Orthogonal polynomials with exponentially decaying recursion coefficients. Proceedings of the conference "Probability and Mathematical Physics" celebrating the 65th birthday of S. Molchanov. CRM Proceedings Series. American Mathematical Society, Providence, R.I., in press. arXiv: math.SP/0603099, 2006.
45. Simon, B. Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle. J. Math. Anal. Appl, in press. arXiv: math.SP/0606037, 2006.
46. Stoiciu, M. Zeros of random orthogonal polynomials on the unit circle. Doctoral dissertation, California Institute of Technology, 2005. Available at: http://resolver.caltech.edu/CaltechETD:etd-05272005-110242
47. Stoiciu, M. The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle. J. Approx. Theory 139 (2006), no. 1-2, 29-64.
48. Szabados, J.; Vértesi, P. Interpolation of functions. World Scientific, Teaneck, N.J., 1990.
49. Szego, G. Über einen Satz des Herrn Serge Bernstein. Schriften Königsberger Gelehrten. Gesellschaft, Naturwissenschaftliche Klasse, 5 (1928), 59-70.
50. Szego, G. Orthogonal polynomials. 3rd ed. American Mathematical Society Colloquium Publications, 23. American Mathematical Society, Providence, R.I., 1967.
51. Vértesi, P. On the zeros of Jacobi polynomials. Studia Sci. Math. Hungar. 25 (1990), no. 4, 401-405.
52. Vértesi, P. On the zeros of generalized Jacobi polynomials. The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin. Ann. Numer. Math. 4 (1997), no. 1-4, 561-577.
53. Vértesi, P. Uniform asymptotics of derivatives of orthogonal polynomials based on generalized Jacobi weights. Acta Math. Hungar. 85 (1999), no. 1-2, 97-130.
54. Weisstein, E. W. Jacobi polynomial. From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/JacobiPolynomial.html
55. Wong, M.-W. L. First and second kind paraorthogonal polynomials and their zeros. Preprint, 2006.