Scattering theory for quasi-one-dimensional tunneling structures

Quantum theoretical studies of semiconductor microstructures are most naturally done in terms of one-dimensional scattering states, which are characterized far from a structure by k-dependent reflection and transmission amplitudes. We have investigated these states using integral forms such as the Lippmann-Schwinger equation. These allow us to obtain global properties of the states and provide the basis for a formal scattering theory of the kind that has been developed for the conventional problem of three-dimensional potential scattering. We find orthonormality relations for the scattering states, the one-dimensional analogues of Wigner's inequality and Levinson's theorem, and associated properties of the complex-momentum transmission amplitude.