Riccati equations and quasi-1D noninteracting problems

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.