The spectrum of Hill's equation

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 \(-\infty <\lambda _{0} <\lambda _{1} \leqq \lambda _{2} <\lambda _{3} \leqq \lambda _{4} < \cdots <\lambda _{2i-1} \leqq \lambda _{2i} \uparrow \infty \). Let the number of simple eigenvalues be \(2n + 1 \leqq \infty \). Borg [1] proved that n = 0 if and only if q is constant. Hochstadt [16] proved that n = 1 if and only if q = c + 2p with a constant c and a Weierstrassian elliptic function p. Lax [22] notes that n = m if q = 4k2K2m(m + 1)sn2(2Kx, k). (sn is the customary elliptic function of Jacobi, K being the complete elliptic integral of modulus k.) The present paper studies the case \(n < \infty \), continuing investigations of Borg [1], Buslaev and Faddeev [2], Dikii [3, 4], Flaschka [8], Gardner et al. [10], Gelfand [11], Gelfand and Levitan [13], Hochstadt [16], and Lax [22, 23] 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 \(\ell(\lambda ) = \sqrt{-(\lambda -\lambda _{0 } )(\lambda -\lambda _{1 } )\cdots (\lambda -\lambda _{2n } )}\). The case \(n = \infty \) 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 [28], Dubrovin and Novikov [6] and A. R. Its and V. B. Matveev [18].