# Application of the spectral element method to the solution of the multichannel Schr\"odinger equation

## TL;DR

This paper demonstrates the application of the spectral element method to accurately solve the multichannel Schrödinger equation, efficiently computing bound states and scattering matrices with fewer grid points.

## Contribution

The paper introduces a spectral element approach for multichannel Schrödinger equations, offering high accuracy and computational efficiency over traditional methods.

## Key findings

- Spectral accuracy achieved with fewer grid points.
- Efficient solution of scattering matrices using sparse matrix solvers.
- Comparison shows advantages over log-derivative propagators.

## Abstract

We apply the spectral element method to the determination of scattering and bound states of the multichannel Schr\"odinger equation. In our approach the reaction coordinate is discretized on a grid of points whereas the internal coordinates are described by either purely diabatic or locally diabatic (diabatic-by-sector) bases. Bound levels and scattering matrix elements are determined with spectral accuracy using relatively small numbers of points. The scattering problem is cast as a linear system solved using state-of-the-art sparse matrix non iterative packages. Boundary conditions can be imposed so to compute a single column of the matrix solution. A comparison with log-derivative propagators customarily used in molecular physics is performed. The same discretization scheme can also be applied to bound levels that are computed using direct scalable sparse-matrix solvers.