# From Schrödinger Spectra to Orthogonal Polynomials Via a Functional Equation

## Abstract

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.

### Keywords

Kato Clarification cosB Lora## Preview

Unable to display preview. Download preview PDF.

### References

- [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]Bellissard J., Simon B. Cantor spectrum for the almost Mathieu equation.
*J. Fund. Anal*.,**48**: 408–419, 1982.MathSciNetMATHCrossRefGoogle Scholar - [3]Berezanski Yu. M. Expansions in Eigenfunctions of Self-Adjoint Operators.
*Amer. Math. Soc*., Providence, RI, 1968.Google Scholar - [4]Chihara T. S.
*An Introduction to Orthogonal Polynomials*. Gordon amp; Breach, New York, 1978.MATHGoogle Scholar - [5]Cycon H. L., Froese R. G., Kirsch W., Simon B.
*Schrödinger Operators*. Springer-Verlag, Berlin, 1987.MATHGoogle Scholar - [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]Gaspar G., Rahman M.
*Basic Hypergeometric Series*. Cambridge University Press, Cambridge, 1990.Google Scholar - [8]Geronimo J. S., van Assche W. Orthogonal polynomials with asymptotically periodic recurrence coefficients.
*J. Approx. Th*.,**46**: 251–283, 1986.MATHCrossRefGoogle Scholar - [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]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]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]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]Hardy G. H., Riesz M.
*The General Theory of Dirichle’s Series*. Cambridge University Press, Cambridge, 1915.Google Scholar - [14]Henrici P.
*Applied and Computational Complex Analysis*, Volume III. Wiley Interscience, New York, 1986.MATHGoogle Scholar - [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]Iserles A. On the generalized pantograph functional-differential equation.
*Europ. J. Appl. Math*.,**4**: 1–38, 1993.MathSciNetMATHGoogle Scholar - [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]Ismail M. E. H., Mulla F. On the generalized Chebyshev polynomials.
*SIAM J. Math. Anal*.,**18**: 243–258, 1987.MathSciNetMATHCrossRefGoogle Scholar - [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]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]Last Y. Zero measure spectrum for the almost Mathieu operator. Tech. Rep., Technion, Haifa, 1993.Google Scholar
- [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]Nikisin E. M. Discrete Sturm-Liouville operators and some problems of function theory.
*J. Soviet Math*.,**35**: 2679–2744, 1986.CrossRefGoogle Scholar - [24]Simon B. Almost periodic Schrödinger operators: A review.
*Adv. Appl. Maths*,**3**: 463–490, 1982.MATHCrossRefGoogle Scholar - [25]Stahl H., Totik V.
*General Orthogonal Polynomials*. Cambridge University Press, Cambridge, 1992.MATHCrossRefGoogle Scholar