Mathematical analysis of a Mu\~noz-Delgado model for cigar-shaped Bose-Einstein condensates

TL;DR

This paper provides a rigorous mathematical analysis of a one-dimensional model for cigar-shaped Bose-Einstein condensates, establishing key properties of ground states and deriving formulas relating energy, chemical potential, and physical parameters.

Contribution

It offers the first detailed mathematical proof of existence, uniqueness, stability, and decay properties of ground states in this model, along with formulas linking energy and particle number.

Findings

Existence and uniqueness of ground states

Orbital stability and Gaussian decay of solutions

Formulas for minimal energy and chemical potential

Abstract

In this paper we present mathematical analysis of one-dimensional effective models proposed in [\cite{MunozDelgado}] concerning Bose-Einstein condensates in the presence of harmonic confinement. Among the demonstrated properties, we can mention: existence, uniqueness, orbital stability, symmetry and gaussian asymptotic decay of ground-state solutions in the repulsive case. We also report formul\ae\ for the minimal energy $E_{\mn}$ and the associate chemical potential $\mu$ as functions of a parameter $\lambda$, which is related to $N$ (the number of atoms) and/or $a$ (the s-wave scattering length). By considering Taylor's development of the non-quadratic therm of the energy and using appropriate gaussian functions as approximations for the ground state, we present some numerical experiments to illustrate our results.