NUMERICALLY STABLE SOLUTION OF COUPLED-CHANNEL EQUATIONS - THE LOCAL REFLECTION MATRIX

The second order linear Schrodinger equation is transformed to a first order nonlinear differential equation for a quantity rho = (iq-1 psi' -psi)/(iq-1 psi' + psi). In a coupled channel problem all quantities occurring in this equation including q (the WKB wave number) are matrices and rho may be called local reflection matrix. This quantity is closely related to the logarithmic derivative of the Schrodinger function but has no singularities in the classically allowed region. In the asymptotic region where the potential is constant the local reflection matrix approaches the physical reflection matrix. In a pure reflection problem (with an infinite potential on one side) this is the full S-matrix, in a transmission problem (with an activation barrier of finite height) unitarity of the S-matrix can be used to determine most quantities of physical interest from rho. While standard logarithmic derivative methods can become instable for transmission problems the solution with the local reflection matrix is completely stable both for reflection and transmission problems.