Some inverse results for Hill's equation

Abstract

We consider Hill's equation y″+( λ− q) y=0 where q∈ L 1[0, π]. We show that if l n —the length of the n-th instability interval—is of order O( n −( k+2) ) then the real Fourier coefficients a n k , b n k 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,….