Inverse problems for the Hill and Dirac operators (Q1594379)

Using the so-called direct method based on a theorem of nonlinear functional analysis the author studies properties of the spectra of the Hill operator \[ T=-\frac{d^2}{dx^2}+q(x) \] in \(L^2({\mathbb{R}})\) with a real 1-periodic potential \(q(x)\in L^2(0,1)\), and the Dirac operator \[ T_D=\left[ \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right] \frac{d}{dx} +\left[ \begin{matrix} q_1(x) & q_2(x) \\ q_2(x) & -q_1(x) \end{matrix} \right] \] in \(L^2({\mathbb{R}})+L^2({\mathbb{R}})\) with real 1-periodic functions \(q_1(x), q_2(x)\in L^2(0,1)\).