The Integral Representation for the Product of Two Parabolic Cylinder Functions $D_\nu (x) D_\nu (-x)$ at $Re \nu <0$ by Means of the Fundamental Solution of a Landau-Type Operator

Abstract

The fundamental solution (Green's function) of a first order matrix ordinary differential equation arising in a Landau-type problem is calculated by two methods. The coincidence of the two representations results in the integral formula for the product of two parabolic cylinder functions $D_\nu(x) D_\nu(-x)$ at $Re \nu <0$, $x$ is real.