Historical remarks to finite-gap integration theory: elementary treatment of the theory

Abstract

In the example of the Schrödinger/KdV equation we give elementary treatment of the theory of finite-gap integration. The concept is equivalent to two kinds of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's integrability of finite-dimensional Hamiltonian systems (stationary KdV--equations). Three key objects in this field: the new explicit formula for the $\Psi$-function, trace formula and the Jacobi problem provide a complete solution. Appendix contains Russian translation of two papers of J.Drach

23 pages; in Russian