# Phase-amplitude formalism for ultra-narrow shape resonances

## TL;DR

This paper introduces a phase-amplitude formalism based on Milne's representation to accurately analyze ultra-narrow shape resonances in quantum scattering, especially in systems with separated classically allowed regions.

## Contribution

The authors develop a novel formalism that employs region-specific amplitude and phase functions to improve the calculation of phase shifts and tunneling rates in complex scattering problems.

## Key findings

- Accurately computes scattering phase shifts for ultra-narrow resonances.
- Provides precise tunneling rates in systems with large barriers.
- Demonstrates effectiveness with an attractive potential example.

## Abstract

We apply Milne's phase-amplitude representation [W. E. Milne, Phys. Rev. 35, 863 (1930)] to a scattering problem involving disjoint classically allowed regions separated by a barrier. Specifically, we develop a formalism employing different sets of amplitude and phase functions --- each set of solutions optimized for a separate region --- and we use these locally adapted solutions to obtain the true value of the scattering phase shift and accurate tunneling rates for ultra-narrow shape resonances. We show results for an illustrative example of an attractive potential with a large centrifugal barrier.