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The Spectrum of Hill’s Equation

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Part of the Contemporary Mathematicians book series (CM)

Abstract

Let q be an infinitely differentiable function of period 1. Then the spectrum of Hill’s operator Q = −d2dx2 + q(x) in the class of functions of period 2 is a discrete series \(-\infty <\lambda _{0} <\lambda _{1} \leqq \lambda _{2} <\lambda _{3} \leqq \lambda _{4} < \cdots <\lambda _{2i-1} \leqq \lambda _{2i} \uparrow \infty \). Let the number of simple eigenvalues be \(2n + 1 \leqq \infty \). Borg [1] proved that n = 0 if and only if q is constant. Hochstadt [16] proved that n = 1 if and only if q = c + 2p with a constant c and a Weierstrassian elliptic function p. Lax [22] notes that n = m if q = 4k2K2m(m + 1)sn2(2Kx, k). (sn is the customary elliptic function of Jacobi, K being the complete elliptic integral of modulus k.) The present paper studies the case \(n < \infty \), continuing investigations of Borg [1], Buslaev and Faddeev [2], Dikii [3, 4], Flaschka [8], Gardner et al. [10], Gelfand [11], Gelfand and Levitan [13], Hochstadt [16], and Lax [22, 23] in various directions. The content may be summed up in the statement that q is an abelian function; in fact, from the present standpoint, the whole subject appears as a part of the classical function theory of the hyperelliptic irrationality \(\ell(\lambda ) = \sqrt{-(\lambda -\lambda _{0 } )(\lambda -\lambda _{1 } )\cdots (\lambda -\lambda _{2n } )}\). The case \(n = \infty \) requires the development of the theory of abelian and theta functions for infinite genus; this will be reported upon in another place. Some of the results have been obtained independently by Novikov [28], Dubrovin and Novikov [6] and A. R. Its and V. B. Matveev [18].

Keywords

Theta Function Trace Formula Symmetric Polynomial Simple Spectrum Double Eigenvalue 
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.

References

  1. [1]
    G. Borg. Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math., 78:1–96, 1946.Google Scholar
  2. [2]
    V. S. Buslaev and L. D. Faddeev. Formulas for traces for a singular Sturm-Liouville differential operator. Dokl. Akad. Nauk SSSR, 132:13–16, 1960.MathSciNetGoogle Scholar
  3. [3]
    E. T. Copson. An Introduction to the Theory of Functions of a Complex Variable. Oxford University Press, Oxford, 1960.Google Scholar
  4. [4]
    L. A. Dikiĭ. The zeta function of an ordinary differential equation on a finite interval. Izvestiya Akad. Nauk SSSR. Ser. Mat., 19:187–200, 1955.Google Scholar
  5. [5]
    L. A. Dikiĭ. Trace formulas for Sturm-Liouville differential operators. Amer. Math. Soc. Trans. (2), 18:81–115, 1961.Google Scholar
  6. [6]
    B. Dubrovin and S. Novikov. Periodic and conditionally periodic analogues of multi-soliton solutions of the Kortweg-de Vries equation. Dokl. Akad. Nauk SSSR, 6:2131–2144, 1974.Google Scholar
  7. [7]
    L. D. Faddeev. The inverse problem in the quantum theory of scattering. Uspehi Mat. Nauk., 14:57–119, 1959. Translation in J. Mathematical Phys. 4:72–104, 1963.Google Scholar
  8. [8]
    H. Flaschka. On the inverse problem for Hill’s operator. Arch. Rational Mech. Anal., 54:293–309, 1975.MathSciNetGoogle Scholar
  9. [9]
    C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura. Method for solving the Korteweg-de Vries equation. Phys. Rev. Letters, 19:1095–1097, 1967.CrossRefGoogle Scholar
  10. [10]
    C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura. Korteweg-de Vries equation and generalizations. VI: methods for exact solution. Comm. Pure Appl. Math., 27:97–133, 1974.Google Scholar
  11. [11]
    I. M. Gelfand. On identities for eigenvalues of a differential operator of second order. Uspehi Mat. Nauk (N.S.), 11:191–198, 1956.Google Scholar
  12. [12]
    I. M. Gelfand and B. M. Levitan. On the determination of a differential equation from its spectral function. Izvestiya Akad. Nauk SSSR. Ser. Mat., 15:309–360, 1951. Translation: American Mathematical Translation, 1 (1955), 253–304.Google Scholar
  13. [13]
    I. M. Gelfand and B. M. Levitan. On a simple identity for the characteristic values of a differential operator of second order. Dokl. Akad. Nauk SSSR, 88:593–596, 1953.MathSciNetGoogle Scholar
  14. [14]
    W. Goldberg. Necessary and sufficient conditions for determining a Hill’s equation from its spectrum. Bull. Amer. Math. Soc., 81:423–424, 1975.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    A. Göpel. Entwurf einer Theorie der Abel’schen Transcendenten erster Ordnung. Ostwald’s Klassiker der exacten Wissenschaften, 67:W. Engelmann, Leipzig, 1895.Google Scholar
  16. [16]
    H. Hochstadt. Function-theoretic properties of the discriminant of Hill’s equation. Math. Zeit., 82:237–242, 1963.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    H. Hochstadt. On the determination of Hill’s equation from its spectrum. Arch. Rat. Mech. Anal., 19:353–362, 1965.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    A. R. Its and V. B. Matveev. Hill operators with a finite number of lacunae. Funkcional. Anal. i Priložen., 9(1):69–70, 1975.MathSciNetCrossRefGoogle Scholar
  19. [19]
    M. Kac. On distributions of certain Wiener functionals. Trans. Amer. Math. Soc., 65:1–13, 1949.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    M. Kac and P. van Moerbeke. On some isospectral second order differential operators. Proc. Nat. Acad. Sci. U.S.A., 71:2350–2351, 1974.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    A. Krazer and W Wirtinger. Abelsche Funktionen und allgemeine Thetafunktionen, volume II,2 of Encyklopädie der Mathematischen Wissenschaften. B. T. Teubner, Leipzig, 1921.Google Scholar
  22. [22]
    P. Lax. Integrals of non-linear equations and solitary waves. Comm. Pure Appl. Math., 21:467–490, 1968.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    P. Lax. Periodic solutions of the KdV equations. In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Postdam, N.Y. 1972), volume 15 of Lectures in Appl. Math,, pages 85–96. Amer. Math. Soc., Rhode Island, 1975.Google Scholar
  24. [24]
    P. Lax. Periodic solutions of the Korteweg-de Vries equation. Comm. Pure Appl. Math., 28:141–188, 1975.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    N. Levinson. The inverse Sturm-Liouville problem. Mat. Tidsskr. B, 1949:25–30, 1949.MathSciNetGoogle Scholar
  26. [26]
    W. Magnus and W. Winkler. Hill’s equation. Interscience Wiley, New York, 1st edition, 1966.zbMATHGoogle Scholar
  27. [27]
    A. Menikoff. The existence of unbounded solutions of the Korteweg-de Vries equation. Comm. Pure Appl. Math., 25:407–432, 1972.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    S. P. Novikov. The periodic problem for the Korteweg-de Vries equation. Funct. Anal. Appl., 8:236–246, 1974.CrossRefzbMATHGoogle Scholar
  29. [29]
    H. E. Rauch and H. M. Farkas. Theta functions with applications to Riemann Surfaces. The Williams & Wilkins Co., Baltimore, 1974.zbMATHGoogle Scholar
  30. [30]
    G. Rosenhain. Abhandlung über die Functionen zweier Variabler mit vier Perioden, welche die Inversen sind der ultra-elliptischen Integrale erster Klasse. Ostwald’s Klassiker der exacten Wissenschaften, 65:W. Engelmann, Leipzig, 1895.Google Scholar
  31. [31]
    C. L. Siegel. Topics in Complex Function Theory II. Automorphic Functions and Abelian Integrals. Wiley-Interscience, 1969.Google Scholar
  32. [32]
    M. J. O. Strutt. Lamesche-, Mathieusche- und verwandte Funktionen in Physik und Technik. Springer-Veralg, 1932.CrossRefGoogle Scholar
  33. [33]
    V. E. Zakharov and L. D. Faddeev. The Korteweg-de Vries equation is a fully integrable Hamiltonian system. Funkcional Anal. i Priložen, 5:18–27, 1974.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Mathematics DepartmentUniversité de LouvainLouvain-la-NeuveBelgium

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