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

Riccati equations and quasi-1D noninteracting problems