2023/05/16 by Peter Gibson, Gibson, Peter · 1 citation
Engineering · Mathematics · #Microwave Imaging and Scattering Analysis #Numerical methods in inverse problems #Geophysical Methods and Applications
paper · pdf · doi:10.48550/arxiv.2305.09505
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.