Schrödinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg–de Vries equation

Explicit description of periodic potentials for which the corresponding Schrodinger operator $N$ possesses only the finite number of energy gaps is obtained. Using this result the solution of the Korteveg–de Vries equation with the “finite-gap” initial condition is expressed, by means of the $N$-dimensional $\Theta$-function, $N$ being the number of the nondegenerate energy gaps. The following characteristic property of the $N$-gap periodic potentials and the $N$-soliton decreasing potentials is discovered: the existence of two solutions $\psi_1(x,\lambda), \psi_2(x,\lambda)$ of the Schrodinger equation, for which the product $\psi_1,\psi_2$ is the polynomial $P$ ($\operatorname{deg}P=N$. $N$ is the number of gaps or the number of bound states of $H$) from the spectral parameter $\lambda$.