Explicit solution of the 1D Schr\"odinger equation

Peter Gibson

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arXiv · 2305.09505

Explicit solution of the 1D Schr\"odinger equation

Peter Gibson

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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 $S$-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.

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Peter Gibson. 2023-05-16. Explicit solution of the 1D Schr\"odinger equation. https://arxiv.org/abs/2305.09505

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