The spectrum of Hill's equation

Letq be an infinitely differentiable function of period 1. Then the spectrum of Hill's operatorQ=?d ²/dx ²+q(x) in the class of functions of period 2 is a discrete series - ∞<λ₀<λ₁≦λ₂<λ₃≦λ₄<...<λ_2i?1≦λ_2i ↑∞. 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 if¹ q=4k ² K ² m(m+1)sn ²(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].