Basis systems on the torus generated by finite-zone integration of the Korteweg-de Vries equation (Q919541)

The existence of a basis system consisting of the eigenfunctions and associated functions of the following non-selfconjugate operator \[ {\mathcal L}=-iV(\partial /\partial \tau)+6(iU(\partial /\partial \tau))f+(iU(\partial /\partial \tau))^ 3 \] in the \(\ell\)-dimensional torus \(T=\{\tau =(\tau_ 1,...,\tau_{\ell})\), \(0\leq \tau_ j\leq 2\pi \}\) is investigated. Here \(f(\tau,E)\in C^{\infty}(T)\)- are the \(\ell\)- zone solutions of the KdV equation, the vectors \(V,U\in {\mathbb{R}}^{2\ell +1}\) have been parametrized by the \(E=(E_ 1,...,E_{2\ell +1})\) points, i.e. by the zone edges of the spectrum of auxiliary Hill operator. Eigenfunctions and associated functions of \({\mathcal L}\) are found by the well-known formulae of finite zone integration theory but the existence of such basis system is not trivial. The main result of the paper consists in the proof of the fact that the system of eigenfunctions of the operator \({\mathcal L}\) is the basis system (in the two zone \(\ell =2\) case). The proof is based on the following observations: i) the eigenfunctions and associated functions of \({\mathcal L}\) depend smoothly on the parameters E; ii) an everywhere dense set exists in the regions of admissible values of parameters E from which the existence of a corresponding basis system follows.