Inverse problem and the trace formula for the Hill operator, II

Abstract.

Consider the Hill operator $T = -d^2/dx^2+q(x)$ on $L^2(\mathbb{R}),$ where $q\in L^2(0,1)$ is a 1-periodic real potential and $\int _0^1q(x)dx=0.$ The spectrum of T is absolutely continuous and consists of intervals separated by gaps $\gamma _n=(a^-_n, a^+_n ), n\geq 1$. Let $\mu_n, n\geq 1,$ be the Dirichlet eigenvalue of the equation $-y''+qy=ły$ on the interval [0,1]. Introduce the vector $g_n=(g_{cn}, g_{sn})\in \mathbb{R}^2,$ with components $g_{cn}=(a_n^++a_n^+)/2-\mu_n,$ and $g_{sn}=||\gamma _n|^2/4-g_{cn}^2|s_n,$ where the sign $s_n=+$ or $s_n=-$ for all $n\geq 1$. Using nonlinear functional analysis in Hilbert spaces we show, that the mapping $g: q\to g(q)=\{g_n \}_1^{\iy }\in \ell^2\oplus \ell^ 2$ is a real analytic isomorphism. In the second part a trace formula for $q\in L^2(0,1)$ is proved.