# Are smooth pseudopotentials a good choice for representing short-range interactions?

## TL;DR

This paper examines the effectiveness of smooth pseudopotentials for modeling short-range interactions in quantum systems, analyzing basis set convergence and scaling relations to improve numerical simulations of ultracold atoms.

## Contribution

It provides a detailed analysis of basis set scaling for Gaussian pseudopotentials and proposes re-scaling methods to enhance convergence efficiency.

## Key findings

- Number of basis functions scales with the fourth power of the pseudopotential length scale
- Re-scaling basis functions reduces scaling to quadratic
- Numerical examples demonstrate practical feasibility for few-fermion systems

## Abstract

When seeking a numerical representation of a quantum-mechanical multiparticle problem it is tempting to replace a singular short-range interaction by a smooth finite-range pseudopotential. Finite basis set expansions, e.g.~in Fock space, are then guaranteed to converge exponentially. The need to faithfully represent the artificial length scale of the pseudopotential, however, places a costly burden on the basis set. Here we discuss scaling relations for the required size of the basis set and demonstrate the basis set convergence on the example of a two-dimensional system of few fermions with short-range $s$-wave interactions in a harmonic trapping potential. In particular we show that the number of harmonic-oscillator basis functions needed to reach a regime of exponential convergence for a Gaussian pseudopotential scales with the fourth power of the pseudopotential length scale, which can be improved to quadratic scaling when the basis functions are re-scaled appropriately. Numerical examples for three fermions with up to a few hundred single-particle basis functions are presented and implications for the feasibility of accurate numerical multi-particle simulations of interacting ultra-cold atom systems are discussed.