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Inventiones mathematicae

, Volume 30, Issue 3, pp 217–274 | Cite as

The spectrum of Hill's equation

  • H. P. McKean
  • P. van Moerbeke
Article

Abstract

Letq be an infinitely differentiable function of period 1. Then the spectrum of Hill's operatorQ=−d2/dx2+q(x) in the class of functions of period 2 is a discrete series - ∞<λ01≦λ23≦λ4<...<λ2i−1≦λ2i↑∞. Let the numer of simple eigenvalues be 2n+1<=∞. Borg [1] proved thatn=0 if and only ifq is constant. Hochstadt [21] proved thatn=1 if and only ifq=c+2p with a constantc and a Weierstrassian elliptic functionp. Lax [29] notes thatn=m if1q=4k2K2m(m+1)sn2(2Kx,k). The present paper studies the casen<∞, continuing investigations of Borg [1], Buslaev and Faddeev [2], Dikii [3, 4], Flaschka [10], Gardneret al. [12], Gelfand [13], Gelfand and Levitan [14], Hochstadt [21], and Lax [28–30] in various directions. The content may be summed up in the statement thatq 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\(\ell (\lambda ) = \sqrt { - (\lambda - \lambda _0 )(\lambda - \lambda _1 )...(\lambda - \lambda _{2n} )} .\) The casen=∞ 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 [34], Dubrovin and Novikov [6] and A. R. Its and V. B. Matveev [22].

Keywords

Differentiable Function Classical Function Function Theory Theta Function Paper Study 
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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Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • H. P. McKean
    • 1
  • P. van Moerbeke
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Mathematics DepartmentUniversité LouvianLouvian-la-NeuveBelgien

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