Explicit solution of the 1D Schr\"odinger equation
Analysis of PDEs
2023-05-17 v1 Mathematical Physics
math.MP
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 Schr\"odinger equation. Explicit formulas for the transmission coefficient and -matrix of the classical one-dimensional Schr\"odinger 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.
Cite
@article{arxiv.2305.09505,
title = {Explicit solution of the 1D Schr\"odinger equation},
author = {Peter Gibson},
journal= {arXiv preprint arXiv:2305.09505},
year = {2023}
}
Comments
28 pages