The spectrum of Hill's equation

H. P. McKean, P. van Moerbeke

Research output: Contribution to journalArticle

Abstract

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
Volume30
Issue number3
DOIs
StatePublished - Oct 1975

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Hill Equation
If and only if
Irrationality
Elliptic function
Theta Functions
Differentiable
Genus
Eigenvalue
Series
Operator

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

McKean, H. P., & van Moerbeke, P. (1975). The spectrum of Hill's equation. Inventiones Mathematicae, 30(3), 217-274. https://doi.org/10.1007/BF01425567

The spectrum of Hill's equation. / McKean, H. P.; van Moerbeke, P.

In: Inventiones Mathematicae, Vol. 30, No. 3, 10.1975, p. 217-274.

Research output: Contribution to journalArticle

McKean, HP & van Moerbeke, P 1975, 'The spectrum of Hill's equation', Inventiones Mathematicae, vol. 30, no. 3, pp. 217-274. https://doi.org/10.1007/BF01425567
McKean, H. P. ; van Moerbeke, P. / The spectrum of Hill's equation. In: Inventiones Mathematicae. 1975 ; Vol. 30, No. 3. pp. 217-274.
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