Inverse problems associated with the hill operator

Let ℓn be the length of the n-th instability interval of the Hill operator Ly = ―y″ + q(x)y. We prove that if ℓn = o(n―2) and the set {(nπ)2: n is even and n > n0} is a subset of the periodic spectrum of the Hill operator, then q = 0 a.e., where n0 is a sufficiently large positive integer such that ℓn < εn―2 for all n > n0(ε) with some ε > 0. A similar result holds for the anti-periodic case. © 2016 Texas State University