The spectrum of Hill's equation

H. P. McKean, P. van Moerbeke

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Let q be an infinitely differentiable function of period 1. Then the spectrum of Hill's operator Q=-d2/dx2+q(x) in the class of functions of period 2 is a discrete series - ∞<λ01≦λ23≦λ4<...<λ2 i-1≦λ2 i↑∞. Let the numer of simple eigenvalues be 2 n+1<=∞. Borg [1] proved that n=0 if and only if q is constant. Hochstadt [21] proved that n=1 if and only if q=c+2 p with a constant c and a Weierstrassian elliptic function p. Lax [29] notes that n=m if1q=4 k2K2m(m+1)sn2(2 Kx,k). The present paper studies the case n<∞, continuing investigations of Borg [1], Buslaev and Faddeev [2], Dikii [3, 4], Flaschka [10], Gardner et 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 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 {Mathematical expression} The case n=∞ 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].

Original languageEnglish (US)
Pages (from-to)217-274
Number of pages58
JournalInventiones Mathematicae
Issue number3
StatePublished - Oct 1975

ASJC Scopus subject areas

  • General Mathematics


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