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Comparative asymptotics of solutions and trace formulas for a class of difference equations

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Abstract

Properties of Jacobi operators generated by Markov functions are studied. The main results refer to the case where the support of the corresponding spectral measure µ consists of several intervals of the real line. In this class of operators, a comparative asymptotic formula for two solutions of the corresponding difference equation, polynomials orthogonal with respect to the measure µ and functions of the second kind (Weyl solutions) is found. Asymptotic trace formulas for the coefficients a n and b n in this difference equation are obtained. The derivation of the asymptotic formulas is based on standard techniques for studying the asymptotic properties of polynomials orthogonal on several intervals and consists in reducing the asymptotic problem to investigating properties of solutions to the Nuttall singular integral equation.

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References

  1. P. L. Chebyshev, “On Continued Fractions,” Uchen. Zap. Imp. Akad. Nauk III, 636–664 (1855).

    Google Scholar 

  2. N. I. Akhiezer, “The Chebyshev Direction in Function Theory,” inMathematics of the XIX Century (Nauka, Moscow, 1987), pp. 9–79 [in Russian].

    Google Scholar 

  3. T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse 8, J1–122 (1894); 9, A1-47 (1895).

    MathSciNet  Google Scholar 

  4. P. Tchébycheff, “Sur le dévelopment des fonctions à une seule variable,” Bull. Acad. Sci. St.-Petersburg Cl. Phys.-Math. 1, 193–200 (1860).

    Google Scholar 

  5. L. D. Faddeev and O. A. Yakubovskii, Lectures on Quantum Mechanics for Student-Mathematicians (Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk, 2001).

    Google Scholar 

  6. G. Szégö, Orthogonal Polynomials (Amer.Math. Soc., New York, 1959; Inostrannaya Literatura, Moscow, 1961).

    MATH  Google Scholar 

  7. E. M. Nikishin, “Sturm-Liouville Discrete Operator and Some Problems of Function Theory,” Tr. Semin. Im. I.G. Petrovskogo 10, 3–77 (1984).

    MATH  MathSciNet  Google Scholar 

  8. Damanik D., Killip R., and Simon B., “Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schrödinger Operators,” arXiv: math.SP/0309206.

  9. K. M. Case, “Orthogonal Polynomials from the Viewpoint of Scattering Theory,” J. Math. Phys. 15(12), 2166–2174 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. S. Geronimo and K. M. Case, “Scattering Theory and Polynomials Orthogonal on the Real Line,” Trans. Amer.Math. Soc. 258(2), 467–494 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Simon, “The Classical Moment Problem as a Self-Adjoint Finite Difference Operator,” Adv. Math. 137, 82–203 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  12. A. I. Aptekarev and E. M. Nikishin, “Scattering Problem for the Discrete Sturm-Liouville Operator,” Mat. Sb. 121(3), 327–358 (1983).

    MathSciNet  Google Scholar 

  13. A. I. Aptekarev, “Asymptotic Properties of Polynomials Orthogonal on a System of Contours and Periodic Motion of Toda Chains,” Mat. Sb. 125(2), 231–258 (1984).

    MathSciNet  Google Scholar 

  14. V. A. Kalyagin, “Hermite-Padé Approximants and Spectral Analysis of Nonsymmetric Difference Operators,” Mat. Sb. 185(6), 79–100 (1994) [Russian Acad. Sci. Sb.Math. 82 (1), 199–216 (1995)].

    Google Scholar 

  15. D. Damanik and B. Simon, “Jost Functions and Jost Solutions for Jacobi Matrices, I. A Necessary and Sufficient Condition for Szegö Asymptotics,” arXiv: math.SP/0502486.

  16. R. Killip and B. Simon, “Sum Rules for Jacobi Matrices and Their Applications to Spectral Theory,” Ann. Math. (2) 158, 253–321 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  17. I. Egorova and L. Golinskii, “Discrete Spectrum for Complex Perturbations of Periodic Jacobi Matrices,” J. Difference Equations Appl. 11(14), 1185–1203 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  18. A. A. Gonchar, “On the Convergence of Padé Approximants for Some Classes of Meromorphic Functions,” Mat. Sb. 97, 607–629 (1975) [Math. USSR Sb. 26, 555–575 (1975)].

    MathSciNet  Google Scholar 

  19. S. P. Suetin, “Spectral Properties of a Class of Discrete Sturm-Liouville Operators,” Usp. Mat. Nauk 61(2), 171–172 (2006) [Russian Math. Surveys 61 (2), 365–367 (2006)].

    MathSciNet  Google Scholar 

  20. A. A. Gonchar, “On the Uniform Convergence of Diagonal Padé Approximants,” Mat. Sb. 118(4), 535–556 (1982).

    MathSciNet  Google Scholar 

  21. E. A. Rakhmanov, “On Ratio Asymptotics for Orthogonal Polynomials,” Mat. Sb. 103(2), 237–252 (1977).

    MathSciNet  Google Scholar 

  22. S. P. Suetin, “Approximation Properties of the Poles of Diagonal Padé Approximants for Certain Generalizations of Markov Functions,” Mat. Sb. 193(12), 81–114 (2002) [Russian Acad. Sci. Mat. Sb. 193 (12), 1837–1866 (2002)].

    MathSciNet  Google Scholar 

  23. A. A. Gonchar and S. P. Suetin, “On Padé Approximants for Markov-Type Meromorphic Functions,” in Modern Problems ofMathematics (Mat. Inst. im.V.A. Steklova, Ross. Akad.Nauk, Moscow, 2004), Vol. 5, pp. 3–65 [in Russian]; Proc. Sleklov Inst.Math. 272, Suppl. 2, 57–94 (2011).

    Google Scholar 

  24. S. P. Suetin, “On the Interpolation Properties of Diagonal Padé Approximants of Elliptic Functions,” Usp. Mat. Nauk 59(4), 201–202 (2004) [RussianMath. Surveys 59 (4), 800–802 (2004)].

    MathSciNet  Google Scholar 

  25. S. P. Suetin, “On the Uniform Convergence of Diagonal Padé Approximants for Hyperelliptic Functions,” Mat. Sb. 191(9), 81–114 (2000) [Russian Acad. Sci. Sb.Math. 191 (9), 1339–1373 (2000)].

    MathSciNet  Google Scholar 

  26. S. P. Suetin, “On the Dynamics of “Wandering” Zeros of Polynomials Orthogonal on Some Segments,” Usp. Mat. Nauk 57(2), 199–200 (2002) [Russian Math. Surv. 57 (2), 425–427 (2002)].

    MathSciNet  Google Scholar 

  27. S. P. Suetin, “Padé Approximants and Efficient Analytic Continuation of a Power Series,” Usp. Mat. Nauk 57(1), 45–142 (2002) [Russian Math. Surveys 57 (1), 43–141 (2002)]. 57 (1), 45–142 (2002).

    MathSciNet  Google Scholar 

  28. J. Nuttall, “Pade Polynomial Asymptotics from a Singular Integral Equation,” Constr. Approx. 6(2), 157–166 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  29. E. A. Rakhmanov, “On the Convergence of Diagonal Padé Approximants,” Mat. Sb. 104(2), 271–291 (1977).

    MathSciNet  Google Scholar 

  30. H. Widom, “Extremal Polynomials Associated with a System of Curves in the Complex Plane,” Adv. Math. 3, 127–232 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  31. E. I. Zverovich, “Boundary Value Problems of the Theory of Analytic Functions in Hölder Classes on Riemann Surfaces,” Usp. Mat. Nauk 26(1), 113–180 (1971).

    MATH  Google Scholar 

  32. G. Springer, Introduction to Riemann Surfaces (Addison-Wesley, Reading, Mass., 1957; Inostrannaya Literatura, Moscow, 1960).

    MATH  Google Scholar 

  33. B. A. Dubrovin, “Theta-Function and Nonlinear Equations,” Usp. Mat. Nauk 36(2), 11–80 (1981).

    MathSciNet  Google Scholar 

  34. N. I. Akhiezer, “Orthogonal Polynomials on Several Intervals,” Dokl. Akad. Nauk SSSR 134(1), 9–12 (1960) [SovietMath. Dokl. 1, 989–992 (1960)].

    MathSciNet  Google Scholar 

  35. N. I. Akhiezer and Yu. Ya. Tomchuk, “On the Theory of Orthogonal Polynomials over Several Intervals,” Dokl. Akad. Nauk SSSR 138(4), 743–746 (1961) [SovietMath. Dokl. 2, 687–690 (1961)].

    MathSciNet  Google Scholar 

  36. N. I. Akhiezer, “A Continuous Analogue of Orthogonal Polynomials on a System of Intervals,” Dokl. Akad. Nauk SSSR 141(2), 263–266 (1961) [SovietMath. Dokl. 2, 1409–1412 (1961)].

    MathSciNet  Google Scholar 

  37. B.M. Levitan, The Inverse Sturm-Liouville Problem (Nauka, Moscow, 1984) [in Russian].

    Google Scholar 

  38. V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1989) [in Russian].

    Google Scholar 

  39. B. A. Dubrovin, “The Periodic Problem for the Korteweg-de Vries Equation in the Class of Finite-Gap Potentials,” Funkts. Anal. Ego Prilozh. 9(3), 41–52 (1975).

    MATH  MathSciNet  Google Scholar 

  40. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable (Nauka, Moscow, 1966) [in Russian].

    MATH  Google Scholar 

  41. J. S. Jeronimo and W. Van Assche, “Orthogonal Polynomials with Asymptotically Periodic Recurrence Coefficients,” J. Approx. Theory 46, 251–283 (1986).

    Article  MathSciNet  Google Scholar 

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  1. Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    S. P. Suetin

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Original Russian Text © S.P. Suetin, 2006, published in Sovremennye Problemy Matematiki, 2006, Vol. 6, pp. 4–74.

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Suetin, S.P. Comparative asymptotics of solutions and trace formulas for a class of difference equations. Proc. Steklov Inst. Math. 272 (Suppl 2), 96–137 (2011). https://doi.org/10.1134/S0081543811030060

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