One-Dimensional Schrödinger Operator on the Half-Line: the Differential Equation for Eigenfunctions with Respect to the Spectral Parameter and an Analog of the Freud Equation

It is shown that the eigenfunctions of the Schrödinger operator on the half-line satisfy an explicitly constructed differential equation with respect to the spectral parameter. Such an equation was earlier obtained for orthogonal polynomials. An analog of the Freud equation is found.