Accurate Complex Scaling of Three Dimensional Numerical Potentials

TL;DR

This paper presents a wavelet-based method for accurately implementing complex scaling in three-dimensional numerical potentials, enabling efficient resonance calculations without artificial parameters.

Contribution

The authors introduce a novel wavelet-based approach for complex scaling on numerical grids, improving accuracy and generality in resonance computations.

Findings

Efficient and accurate complex scaling in 3D potentials.

Wavelet basis set eliminates artificial convergence parameters.

Potential to advance computational non-Hermitian quantum mechanics.

Abstract

The complex scaling method, which consists in continuing spatial coordinates into the complex plane, is a well-established method that allows to compute resonant eigenfunctions of the time-independent Schroedinger operator. Whenever it is desirable to apply the complex scaling to investigate resonances in physical systems defined on numerical discrete grids, the most direct approach relies on the application of a similarity transformation to the original, unscaled Hamiltonian. We show that such an approach can be conveniently implemented in the Daubechies wavelet basis set, featuring a very promising level of generality, high accuracy, and no need for artificial convergence parameters. Complex scaling of three dimensional numerical potentials can be efficiently and accurately performed. By carrying out an illustrative resonant state computation in the case of a one-dimensional model potential, we then show that our wavelet-based approach may disclose new exciting opportunities in the field of computational non-Hermitian quantum mechanics.