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Finite Gap Jacobi Matrices, I. The Isospectral Torus

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Constructive Approximation Aims and scope

Abstract

Let \(\frak{e}\subset\mathbb{R}\) be a finite union of disjoint closed intervals. In the study of orthogonal polynomials on the real line with measures whose essential support is  \(\frak{e}\) , a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent presentation of properties and bounds for this special class as a tool for ourselves and others to study perturbations. One important result is the expression of Jost functions for the torus in terms of theta functions.

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References

  1. Ahlfors, L.V.: Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable. McGraw–Hill, New York (1978)

    Google Scholar 

  2. Ahlfors, L., Bers, L.: Riemann’s mapping theorem for variable metrics. Ann. Math. (2) 72, 385–404 (1960)

    Article  MathSciNet  Google Scholar 

  3. Aptekarev, A.I.: Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains. Math. USSR Sb. 53, 233–260 (1986). Russian original in Mat. Sb. (N.S.) 125(167), 231–258 (1984)

    Article  MATH  Google Scholar 

  4. Armstrong, M.A.: Basic Topology. Undergraduate Texts in Mathematics. Springer, New York (1983). Corrected reprint of the 1979 original

    MATH  Google Scholar 

  5. Avila, A., Last, Y., Simon, B.: Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with a.c. spectra, preprint

  6. Barrios Rolanía, D., López Lagomasino, G.: Ratio asymptotics for polynomials orthogonal on arcs of the unit circle. Constr. Approx. 15, 1–31 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Batchenko, V., Gesztesy, F.: On the spectrum of Jacobi operators with quasi-periodic algebro-geometric coefficients. Int. Math. Res. Pap. 10, 511–563 (2005)

    Article  MathSciNet  Google Scholar 

  8. Beardon, A.F.: Inequalities for certain Fuchsian groups. Acta Math. 127, 221–258 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bello Hernández, M., López Lagomasino, G.: Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle. J. Approx. Theory 92, 216–244 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Borthwick, D.: Spectral Theory of Infinite-Area Hyperbolic Surfaces. Progress in Mathematics, vol. 256. Birkhäuser Boston, Boston (2007)

    Google Scholar 

  11. Breuer, J., Last, Y., Simon, B.: The Nevai condition. Constr. Approx., to appear

  12. Bulla, W., Gesztesy, F., Holden, H., Teschl, G.: Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac–van Moerbeke hierarchies. Memoirs Am. Math. Soc. 135(641) (1998)

  13. Burnside, W.: On a class of automorphic functions. Proc. Lond. Math. Soc. 23, 49–88 (1891)

    Article  Google Scholar 

  14. Burnside, W.: Further note on automorphic functions. Proc. Lond. Math. Soc. 23, 281–295 (1891)

    Article  Google Scholar 

  15. Christiansen, J., Simon, B., Zinchenko, M.: Finite gap Jacobi matrices: An announcement. J. Comput. Appl. Math., to appear

  16. Christiansen, J., Simon, B., Zinchenko, M.: Finite gap Jacobi matrices, II. The Szegő class, in preparation

  17. Christiansen, J., Simon, B., Zinchenko, M.: Finite gap Jacobi matrices, III. Beyond the Szegő class, in preparation

  18. Craig, W.: The trace formula for Schrödinger operators on the line. Commun. Math. Phys. 126, 379–407 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  19. Damanik, D., Simon, B.: Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics. Invent. Math. 165, 1–50 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Damanik, D., Killip, R., Simon, B.: Perturbations of orthogonal polynomials with periodic recursion coefficients. Ann. Math., to appear

  21. Denissov, S.A.: On Rakhmanov’s theorem for Jacobi matrices. Proc. Am. Math. Soc. 132, 847–852 (2004)

    Article  Google Scholar 

  22. Dubrovin, B.A., Matveev, V.B., Novikov, S.P.: Nonlinear equations of Korteweg–de Vries type, finite-band linear operators and Abelian varieties. Usp. Mat. Nauk 31(1(187)), 55–136 (1976). (Russian)

    MATH  MathSciNet  Google Scholar 

  23. Erdös, P., Turán, P.: On interpolation. III. Interpolatory theory of polynomials. Ann. Math. (2) 41, 510–553 (1940)

    Article  Google Scholar 

  24. Farkas, H.M., Kra, I.: Riemann Surfaces. Graduate Texts in Mathematics, vol. 71. Springer, New York (1980)

    MATH  Google Scholar 

  25. Flaschka, H., McLaughlin, D.W.: Canonically conjugate variables for the Korteweg–de Vries equation and the Toda lattice with periodic boundary conditions. Progr. Theor. Phys. 55, 438–456 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  26. Fonseca, I., Gangbo, W.: Degree Theory in Analysis and Applications. Oxford Lecture Series in Mathematics and Its Applications, vol. 2. Clarendon Press/Oxford University Press, New York (1995)

    MATH  Google Scholar 

  27. Frank, R., Simon, B., Weidl, T.: Eigenvalue bounds for perturbations of Schrödinger operators and Jacobi matrices with regular ground states. Commun. Math. Phys. 282, 199–208 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Freud, G.: Orthogonal Polynomials. Pergamon, Oxford (1971)

    Google Scholar 

  29. Gesztesy, F., Holden, H.: Soliton Equations and Their Algebro-Geometric Solutions. Vol. I: (1+1)-Dimensional Continuous Models. Cambridge Studies in Advanced Mathematics, vol. 79. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  30. Gesztesy, F., Holden, H., Michor, J., Teschl, G.: Soliton Equations and Their Algebro-Geometric Solutions. Vol. II: (1+1)-Dimensional Discrete Models. Cambridge Studies in Advanced Mathematics, vol. 114. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  31. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)

    MATH  Google Scholar 

  32. Guillemin, V., Pollack, A.: Differential Topology. Prentice–Hall, Englewood Cliffs (1974)

    MATH  Google Scholar 

  33. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  34. Hejhal, D.A.: Universal covering maps for variable regions. Math. Z. 137, 7–20 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  35. Helms, L.L.: Introduction to Potential Theory. Pure and Applied Mathematics, vol. 22. Wiley–Interscience, New York (1969)

    MATH  Google Scholar 

  36. Hundertmark, D., Simon, B.: Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices. J. Math. Anal. Appl. 340, 892–900 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  37. Killip, R., Simon, B.: Sum rules for Jacobi matrices and their applications to spectral theory. Ann. Math. 158, 253–321 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  38. Katok, S.: Fuchsian Groups. University of Chicago Press, Chicago (1992)

    MATH  Google Scholar 

  39. Krawcewicz, W., Wu, J.: Theory of Degrees With Applications to Bifurcations and Differential Equations. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1997)

    MATH  Google Scholar 

  40. Krichever, I.M.: Algebraic curves and nonlinear difference equations. Usp. Mat. Nauk 33(4(202)), 215–216 (1978). (Russian)

    MATH  MathSciNet  Google Scholar 

  41. Landkof, N.S.: Foundations of Modern Potential Theory. Springer, Berlin (1972)

    MATH  Google Scholar 

  42. Last, Y., Simon, B.: The essential spectrum of Schrödinger, Jacobi, and CMV operators. J. Anal. Math. 98, 183–220 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  43. Lloyd, N.G.: Degree Theory. Cambridge Tracts in Mathematics, vol. 73. Cambridge University Press, Cambridge (1978)

    MATH  Google Scholar 

  44. Martínez-Finkelshtein, A.: Equilibrium problems of potential theory in the complex plane. In: Orthogonal Polynomials and Special Functions. Lecture Notes in Mathematics, vol. 1883, pp. 79–117. Springer, Berlin (2006)

    Chapter  Google Scholar 

  45. McKean, H.P., van Moerbeke, P.: The spectrum of Hill’s equation. Invent. Math. 30, 217–274 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  46. Milnor, J.W.: Topology From the Differentiable Viewpoint. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997). Revised reprint of the 1965 original

    Google Scholar 

  47. Miranda, R.: Algebraic Curves and Riemann Surfaces. Graduate Studies in Mathematics, vol. 5. American Mathematical Society, Providence (1995)

    MATH  Google Scholar 

  48. Nevai, P.: Orthogonal polynomials. Mem. Am. Math. Soc. 18(213), 1–183 (1979)

    MathSciNet  Google Scholar 

  49. Patterson, S.J.: The limit set of a Fuchsian group. Acta Math. 136, 241–273 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  50. Peherstorfer, F., Yuditskii, P.: Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points. Proc. Am. Math. Soc. 129, 3213–3220 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  51. Peherstorfer, F., Yuditskii, P.: Asymptotic behavior of polynomials orthonormal on a homogeneous set. J. Anal. Math. 89, 113–154 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  52. Peherstorfer, F., Yuditskii, P.: Remark on the paper “Asymptotic behavior of polynomials orthonormal on a homogeneous set”. arXiv:math.SP/0611856

  53. Poincaré, H.: Mémoire sur les fonctions fuchsiennes. Acta Math. 1, 193–294 (1882)

    Article  MathSciNet  Google Scholar 

  54. Radó, T.: Über die Fundamentalabbildungen schlichter Gebiete. Acta Litt. Sci. Univ. Hung. 1, 240–251 (1923)

    Google Scholar 

  55. Ransford, T.: Potential Theory in the Complex Plane. Press Syndicate of the University of Cambridge, New York (1995)

    Book  MATH  Google Scholar 

  56. Remling, C.: The absolutely continuous spectrum of Jacobi matrices, preprint

  57. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw–Hill, New York (1987)

    MATH  Google Scholar 

  58. Simon, B.: A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices. J. Funct. Anal. 214, 396–409 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  59. Simon, B.: Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line. J. Approx. Theory 126, 198–217 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  60. Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory. AMS Colloquium Publications, vol. 54.1. American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  61. Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory. AMS Colloquium Publications, vol. 54.2. American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  62. Simon, B.: Equilibrium measures and capacities in spectral theory. Inverse Probl. Imaging 1, 713–772 (2007)

    MATH  MathSciNet  Google Scholar 

  63. Simon, B.: The Christoffel–Darboux kernel. In: Perspectives in PDE, Harmonic Analysis and Applications. Proc. Sympos. Pure Math., vol. 79, pp. 295–335. American Mathematical Society, Providence (2008). A volume in honor of V.G. Maz’ya’s 70th birthday

    Google Scholar 

  64. Simon, B.: Two extensions of Lubinsky’s universality theorem. J. Anal. Math. 105, 345–362 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  65. Simon, B.: Regularity and the Cesàro–Nevai class. J. Approx. Theory 156, 142–153 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  66. Simon, B.: Szegő’s Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials, in preparation; to be published by Princeton University Press

  67. Sodin, M., Yuditskii, P.: Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal. 7, 387–435 (1997)

    MATH  MathSciNet  Google Scholar 

  68. Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. I, 2nd edn. Publish or Perish, Wilmington (1979)

    Google Scholar 

  69. Stahl, H., Totik, V.: General Orthogonal Polynomials. In: Encyclopedia of Mathematics and Its Applications, vol. 43. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  70. Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math. 50, 171–202 (1979)

    Article  MATH  Google Scholar 

  71. Szegő, G.: Orthogonal Polynomials. AMS Colloquium Publications, vol. 23. American Mathematical Society, Providence (1939); 3rd edn. 1967

    Google Scholar 

  72. Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs, vol. 72. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  73. Totik, V.: Polynomial inverse images and polynomial inequalities. Acta Math. 187, 139–160 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  74. Tsuji, M.: Potential Theory in Modern Function Theory. Chelsea, New York (1975). Reprinting of the 1959 original

    MATH  Google Scholar 

  75. Ullman, J.L.: On the regular behaviour of orthogonal polynomials. Proc. Lond. Math. Soc. (3) 24, 119–148 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  76. Ullman, J.L.: Orthogonal polynomials for general measures. I. In: Rational Approximation and Interpolation, Tampa, FL, 1983. Lecture Notes in Mathematics, vol. 1105, pp. 524–528. Springer, Berlin (1984)

    Chapter  Google Scholar 

  77. Ullman, J.L.: Orthogonal polynomials for general measures. II. In: Orthogonal Polynomials and Applications, Bar-le-Duc, 1984. Lecture Notes in Mathematics, vol. 1171, pp. 247–254. Springer, Berlin (1985)

    Google Scholar 

  78. Van Assche, W.: Invariant zero behaviour for orthogonal polynomials on compact sets of the real line. Bull. Soc. Math. Belg. Ser. B 38, 1–13 (1986)

    MATH  MathSciNet  Google Scholar 

  79. van Moerbeke, P.: The spectrum of Jacobi matrices. Invent. Math. 37, 45–81 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  80. Vassiliev, V.A.: Introduction to Topology. Student Mathematical Library, vol. 14. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  81. Widom, H.: Polynomials associated with measures in the complex plane. J. Math. Mech. 16, 997–1013 (1967)

    MATH  MathSciNet  Google Scholar 

  82. Widom, H.: Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3, 127–232 (1969)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics 253-37, California Institute of Technology, Pasadena, CA, 91125, USA

    Jacob S. Christiansen, Barry Simon & Maxim Zinchenko

  2. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Jacob S. Christiansen

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  1. Jacob S. Christiansen
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  2. Barry Simon
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  3. Maxim Zinchenko
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Correspondence to Barry Simon.

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Communicated by Vilmos Totik.

The work of the second author was supported in part by NSF grant DMS-0652919.

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Christiansen, J.S., Simon, B. & Zinchenko, M. Finite Gap Jacobi Matrices, I. The Isospectral Torus. Constr Approx 32, 1–65 (2010). https://doi.org/10.1007/s00365-009-9057-z

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