Gap estimates of the spectrum of Hill's equation and action variables for KdV

Consider the Schrodinger equation -y" + Vy = lambda y for a potential V of period 1 in the weighted Sobolev space (N is an element of Z(greater than o r equal to 0), omega is an element of R (greater than or equal to 0)) [GRAPHICS] where (f) over cap(k) (k is an element of Z) denote the Fourier coefficient s of f when considered as a function of period 1, [GRAPHICS] and where S-1 is the circle of length 1. Denote by lambda(k) = lambda(k)(V) (k greater than or equal to 0) the periodic eigenvalues of -d(2)/dx(2) + V when considered on the interval [0, 2], with multiplicities and ordered so that Re lambda(j) less than or equal to Re lambda(j+1) (j greater than or equal to 0). We prove the following result. Theorem. For any bounded set B subset of or equal to H (N,omega) (S-1;C), t here exist n(0) greater than or equal to 1 and M greater than or equal to 1 so that for k greater than or equal to n(0) and V is an element of B, the eigenvalues lambda(2k,) lambda(2k-1) are isolated pairs, satisfying (with { lambda(2k), lambda(2k-1)} = {lambda(k)(+), lambda(k)(-)}) [GRAPHICS]