  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.
