Some inverse results for Hill’s equation

We consider Hill’s equation y + (λ − q)y = 0 where q ∈ L1[0,π]. We show that if ln—the length of the n-th instability interval—is of order O(n−(k+2)) then the real Fourier coefficients akn ,bkn of q(k)—k-th derivative of q—are of order O(n−2), which implies that q(k) is absolutely continuous almost everywhere for k = 0, 1, 2, . . . .  2002 Elsevier Science (USA). All rights reserved