The spectrum of Hill's equation

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<...<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 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].