SL(3,C) structure of one-dimensional Schr\"odinger equation

Abstract

We present a new formalism for describing solutions of the one-dimensional stationary Schr\"odinger equation in terms of the Lie group SL(3,C) and its Lie algebra. In this formalism, we obtain a universal expression for the Green function which can be used in any representation of SL(3,C) and also expressions for various quantities involving products of Green functions. Specifically, we introduce an infinite-dimensional representation of SL(3,C) that provides a natural description of multiple scattering of waves. Using this particular representation, we can derive formulas which are useful for the analysis of the Green function.