From Schrödinger Spectra to Orthogonal Polynomials Via a Functional Equation

  • Arieh Iserles
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


The main difference between certain spectral problems for linear Schrödinger operators, e.g. the almost Mathieu equation, and three-term recurrence relations for orthogonal polynomials is that in the former the index ranges across ℤ and in the latter only across ℤ+. We present a technique that, by a mixture of Dirichlet and Taylor expansions, translates the almost Mathieu equation and its generalizations to three term recurrence relations. This opens up the possibility of exploiting the full power of the theory of orthogonal polynomials in the analysis of Schrödinger spectra.

Aforementioned three-term recurrence relations share the property that their coefficients are almost periodic. In the special case when they form a periodic sequence, the support can be explicitly identified by a technique due to Geronimus. The more difficult problem, when the recurrence is almost periodic but fails to be periodic, is still open and we report partial results.

The main promise of the technique of this article is that it can be extended to deal with multivariate extensions of the almost Mathieu equations. However, important theoretical questions need be answered before it can be implemented to the analysis of the spectral problem for Schrödinger operators.


Orthogonal Polynomial Spectral Problem Chebyshev Polynomial Borel Measure Essential Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Avron J., Simon B. Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. Amer. Math. Soc., 6: 81–85, 1982.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Bellissard J., Simon B. Cantor spectrum for the almost Mathieu equation. J. Fund. Anal., 48: 408–419, 1982.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Berezanski Yu. M. Expansions in Eigenfunctions of Self-Adjoint Operators. Amer. Math. Soc., Providence, RI, 1968.Google Scholar
  4. [4]
    Chihara T. S. An Introduction to Orthogonal Polynomials. Gordon amp; Breach, New York, 1978.MATHGoogle Scholar
  5. [5]
    Cycon H. L., Froese R. G., Kirsch W., Simon B. Schrödinger Operators. Springer-Verlag, Berlin, 1987.MATHGoogle Scholar
  6. [6]
    Derfel G. A., Molchanov S. A. Spectral methods in the theory of differential-functional equations. MatematichesJce Zametki Akad. Nauk USSR, No. 3 47: 42–51, 1990.MathSciNetGoogle Scholar
  7. [7]
    Gaspar G., Rahman M. Basic Hypergeometric Series. Cambridge University Press, Cambridge, 1990.Google Scholar
  8. [8]
    Geronimo J. S., van Assche W. Orthogonal polynomials with asymptotically periodic recurrence coefficients. J. Approx. Th., 46: 251–283, 1986.MATHCrossRefGoogle Scholar
  9. [9]
    Geronimo J. S., van Assche W. Orthogonal polynomials on several intervals via a polynomial mapping. Transactions Amer. Math. Soc., 308: 559–581, 1988.MATHCrossRefGoogle Scholar
  10. [10]
    Geronimus Yu. L. On some finite difference equations and corresponding systems of orthogonal polynomials. Zap. Mat. Otd. Fiz.-Mat. Fak. i Kharkov Mat. Obsc., 25 (4): 87–100, 1957.Google Scholar
  11. [11]
    Gesztesy F., Holden H., Simon B., Zhao Z. Trace formulae and inverse spectral theory for Schrödinger operators. Bull. Amer. Math. Soc., 29: 250–255, 1993.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Grosjean C. C. The measure induced by orthogonal polynomials satisfying a recurrence formula with either constant or periodic coefficients. Part II: Pure or mixed periodic coefficients. Acad. Koninkl. Acad. Wetensch. Lett. Sch. Kunsten Belgie, 48 (5): 55–94, 1986.MathSciNetGoogle Scholar
  13. [13]
    Hardy G. H., Riesz M. The General Theory of Dirichle’s Series. Cambridge University Press, Cambridge, 1915.Google Scholar
  14. [14]
    Henrici P. Applied and Computational Complex Analysis, Volume III. Wiley Interscience, New York, 1986.MATHGoogle Scholar
  15. [15]
    Hofstadter D. R. Energy levels and wave functions of Bloch electrons in a rational or irrational magnetic field. Phys. Rev. B, 14: 2239–2249, 1976.CrossRefGoogle Scholar
  16. [16]
    Iserles A. On the generalized pantograph functional-differential equation. Europ. J. Appl. Math., 4: 1–38, 1993.MathSciNetMATHGoogle Scholar
  17. [17]
    Iserles A., Liu Y. On pantograph integro-differential equations. DAMTP Tech. Rep. 1993/NA13, University of Cambridge 1993. To appear in J. Integral Eqns amp; Appls.Google Scholar
  18. [18]
    Ismail M. E. H., Mulla F. On the generalized Chebyshev polynomials. SIAM J. Math. Anal., 18: 243–258, 1987.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Kato T., McLeod B. The functional-differential equation y′(t) = ay(λt) + by(t). Bull. Amer. Math. Soc., 77: 891–937, 1971.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Last Y. A relation between a. c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Comm. Math. Phys., 155: 183–192, 1993.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Last Y. Zero measure spectrum for the almost Mathieu operator. Tech. Rep., Technion, Haifa, 1993.Google Scholar
  22. [22]
    Máté A., Nevai P., van Assche W. The supports of measures associated with orthogonal polynomials and the spectra of the related self-adjoint operators. Rocky Mntn J. Maths, 21: 501–527, 1991.MATHCrossRefGoogle Scholar
  23. [23]
    Nikisin E. M. Discrete Sturm-Liouville operators and some problems of function theory. J. Soviet Math., 35: 2679–2744, 1986.CrossRefGoogle Scholar
  24. [24]
    Simon B. Almost periodic Schrödinger operators: A review. Adv. Appl. Maths, 3: 463–490, 1982.MATHCrossRefGoogle Scholar
  25. [25]
    Stahl H., Totik V. General Orthogonal Polynomials. Cambridge University Press, Cambridge, 1992.MATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • Arieh Iserles
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeEngland

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