Transformation of the Eilenberger Equations of Superconductivity to a Scalar Riccati Equation

A new parametrisation of the Eilenberger equations of superconductivity in terms of the solutions to a scalar differential equation of the Riccati type is introduced. It is shown that the quasiclassical propagator, and in particular the local density of states, may be reconstructed, without explicit knowledge of any eigenfunctions and eigenvalues, by solving a simple initial value problem for the linearised Bogoliubov-de Gennes equations. The Riccati parametrisation of the quasiclassical propagator leads to a stable and fast numerical method to solve the Eilenberger equations. For some spatially varying model pair potentials exact solutions to the Eilenberger Equations are found.