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

TL;DR

This paper introduces a novel formalism using the Lie group SL(3,C) to analyze solutions of the one-dimensional Schrödinger equation, providing universal expressions for Green functions and a new perspective on wave scattering.

Contribution

It develops a new SL(3,C)-based formalism for the Schrödinger equation, including a universal Green function expression and an infinite-dimensional representation for wave scattering analysis.

Findings

Universal Green function expression for SL(3,C) representations

New formulas for products of Green functions

Application of infinite-dimensional representation to wave scattering

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.