Explicit solution of the 1D Schrödinger equation

Abstract

Evaluation of a product integral with values in the Lie group SU(1,1) yields the explicit solution to the impedance form of the Schrödinger equation. Explicit formulas for the transmission coefficient and $S$-matrix of the classical one-dimensional Schrödinger operator with arbitrary compactly supported potential are obtained as a consequence. The formulas involve operator theoretic analogues of the standard hyperbolic functions, and provide a new window on acoustic and quantum scattering in one dimension.

28 pages