Inventiones mathematicae

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

The spectrum of Hill's equation

  • H. P. McKean
  • P. van Moerbeke


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].


Differentiable Function Classical Function Function Theory Theta Function Paper Study 
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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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