Numerical solution of the Schrödinger equation with a non-local exchange kernel
This paper describes a numerical method for the solution of an integral form of Schrödinger's equation suitable for scattering problems. Examples are given of the scattering of a slow electron by a Yukawa potential and also by hydrogenic ions. The method essentially consists of discretization of the integral equation \[ u(r)=f(Kr)- \int^{\infty}_{0}G(r,x)V(x)u(x)x^ 2 dx-\int^{\infty}_{0}x^ 2u(x)\int^{\infty}_{0}G(r,y)W(y,x)y^ 2 dy dx \] so that the points used include those which are used in the numerical integration formulae for the integrals involving u. Another problem is that G and W have discontinuous derivatives and are changing rapidly. However high order Newton-Cotes formulae are used to deal with this problem in a form such as \[ \int^{x_ 0+h}_{x_ 0}y dx=Ay_ 0+By_ 1+Cy_ 2+Dy_ 3+Ey_ 4 \] where the values \(y_ 2,\) \(y_ 3\) and \(y_ 4\) are the analytic continuation of the integrand rather than the values of the integrand.
scientific article; zbMATH DE number 4005566