### Abstract

Let q be an infinitely differentiable function of period 1. Then the spectrum of Hill's operator Q=-d^{2}/dx^{2}+q(x) in the class of functions of period 2 is a discrete series - ∞<λ_{0}<λ_{1}≦λ_{2}<λ_{3}≦λ_{4}<...<λ_{2 i-1}≦λ_{2 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 if^{1}q=4 k^{2}K^{2}m(m+1)sn^{2}(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 language | English (US) |
---|---|

Pages (from-to) | 217-274 |

Number of pages | 58 |

Journal | Inventiones Mathematicae |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - Oct 1975 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Inventiones Mathematicae*,

*30*(3), 217-274. https://doi.org/10.1007/BF01425567

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

Research output: Contribution to journal › Article

*Inventiones Mathematicae*, vol. 30, no. 3, pp. 217-274. https://doi.org/10.1007/BF01425567

}

TY - JOUR

T1 - The spectrum of Hill's equation

AU - McKean, H. P.

AU - van Moerbeke, P.

PY - 1975/10

Y1 - 1975/10

N2 - 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 - ∞<λ0<λ1≦λ2<λ3≦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].

AB - 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 - ∞<λ0<λ1≦λ2<λ3≦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].

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U2 - 10.1007/BF01425567

DO - 10.1007/BF01425567

M3 - Article

VL - 30

SP - 217

EP - 274

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 3

ER -