Riccati equations and quasi-1D noninteracting problems

TL;DR

This paper explores Riccati equations and transfer matrix methods for 1D Schrödinger equations, with applications to superconductivity and free energy calculations, unifying classical boundary value problem results.

Contribution

It introduces a unified approach linking Riccati equations, transfer matrices, and quasiclassical equations for 1D Schrödinger problems, with applications to superconductivity.

Findings

Derived expressions for local Green functions using Riccati equations.

Connected transfer matrix approach to Eilenberger quasiclassical equations.

Presented a gradient expansion for free energy density in superconducting systems.

Abstract

We consider a general 1D matrix Schr\"odinger equation within a transfer matrix approach. For a quadratic kinetic term we discuss expressions for the local Green function in terms of solutions of equations of the Riccati type, and an associated formula for the operator determinant. For a linear kinetic term, the approach reduces to Eilenberger quasiclassical equations. In general, it derives from classical results in boundary value problems. We consider applications to illustrative problems, concentrating on superconductivity, and discuss a general gradient expansion for the free energy density.