The Weber equation as a normal form with applications to top of the barrier scattering

TL;DR

This paper reduces the semiclassical Schrödinger equation near a potential maximum to a Weber normal form using the Liouville-Green transform, enabling precise analysis of top of the barrier scattering.

Contribution

It introduces a method to transform the Schrödinger equation into Weber normal form with regularity-preserving diffeomorphisms, and applies this to accurately analyze scattering near the potential maximum.

Findings

Derived a Weber normal form for the Schrödinger equation near the potential maximum.

Provided an accurate representation of the scattering matrix near the top of the barrier.

Showed the diffeomorphism's regularity matches that of the potential, ensuring robustness.

Abstract

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 Schr\"odinger 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).