2015/10/18 by Rodica D. Costin, Costin, Rodica D., Hyejin Park +3
Mathematics · Physics and Astronomy · #34E20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum optics and atomic interactions #Spectral Theory in Mathematical Physics #math-ph #math.CA #math.MP #msc:34E20
paper · pdf · doi:10.48550/arxiv.1510.05322
arxiv created 2015/10/18 · openalex publication_date 2015/10/18 · arxiv updated 2015/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the paper we revisit the basic problem of tunneling near a nondegenerate global maximum of a potential on the line. We reduce the semiclassical Schrödinger equation to a Weber normal form by means of the Liouville-Green transform. We show that the diffeomorphism which effects this stretching of the independent variable lies in the same regularity class as the potential (analytic or infinitely differentiable) with respect to both variables, i.e., space and energy. We then apply the Weber normal form to the scattering problem for energies near the potential maximum. In particular we obtain a representation of the scattering matrix which is accurate up to multiplicative factors of the form 1 + o(1).