Inverse Problem for Periodic “Weighted” Operators

Define the periodic weighted operator Ty=−ρ−2(ρ2y′)′ in L2(R, ρ(x)2 dx). Suppose a function ρ∈W21(R/Z) is 1-periodic real positive, ρ(0)=1, and let q=ρ′/ρ∈L2(0, 1). The spectrum of T consists of intervals σn=[λ+n−1, λ−n] separated by gaps γn=(λ−n, λ+n), n⩾1, with the lengths |γn| and λ+0=0. Let m2n, n⩾1, be the Dirichlet eigenvalue of the equation −y″−2qy′=z2y, y(0)=y(1)=0 where mn>0. Introduce the Lyapunov function Δ(z, q) for T and note that Δz(zn, q)=0 for some zn∈[λ−n, λ+n]. Let ϕ(x, z, q) be the solution of the equation −ϕ″−2qϕ′=z2ϕ, z∈C, satisfying ϕ(0, z, q)=0, ϕ′(0, z, q)=1. Introduce the vector hn=(hcn, hsn)∈R2, with components hcn=−log[(−1)n ϕ′(1, mn, q)], hsn=||hn|2−(hcn)2|1/2 sign(zn−mn), where |hn| is defined by the equation cosh |hn|=(−1)n Δ(zn, q)⩾1 and coincides with the euclidien norm of the vector hn. Using nonlinear functional analysis in Hilbert space, we prove that the mapping h: q→h(q)={hn}∞1 is a real analytic isomorphism.