Invariance principle for inverse problems

Consider the nonlinear mapping F : H2 → H given by F(y) = y″+u1(y′)+u2(y)−u0(y,y′), y ∈ H2, where the Hilbert spaces $H={\{q:q\in L_{\text{?}R}^{\text{?}2}{(\text{T})},\int_{\text{?}0}^{\text{?}1}q{(x)}\text{?}dx=0\}}$ and H2 = {y,y″ ∈ H}. Here, the functions u1, u2 are real analytic and u2′(y)≤ 0; the constant u0 is such that F(y) ∈ H. We prove that the map F is a real analytic isomorphism. Furthermore, a priori 2-sided estimates of the norms of F(y), y are obtained. We apply these results to the inverse problems for the Schrödinger operators −d²dx²+q(x) in L²(ℝ) with a 1-periodic potential. In particular, we use this result to study the Camassa-Holm equation.