On the Thomas-Fermi approximation of the ground state in a PT-symmetric confining potential

TL;DR

This paper analyzes the ground state of the one-dimensional Gross-Pitaevskii equation with PT-symmetric harmonic and linear imaginary potentials, demonstrating the validity of the Thomas-Fermi approximation at large densities through mathematical and numerical methods.

Contribution

It provides a rigorous construction and justification of the Thomas-Fermi approximation for PT-symmetric potentials using dynamical systems and Painlevé-II reduction, including numerical validation.

Findings

Thomas-Fermi approximation constructed via invertible coordinate transformation.

Existence problem reduced to Painlevé-II equation with Hastings-McLeod solution.

Numerical iterative approach converges, but lacks analytical proof.

Abstract

For the stationary Gross-Pitaevskii equation with harmonic real and linear imaginary potentials in the space of one dimension, we study the ground state in the limit of large densities (large chemical potentials), where the solution degenerates into a compact Thomas-Fermi approximation. We prove that the Thomas-Fermi approximation can be constructed with an invertible coordinate transformation and an unstable manifold theorem for a planar dynamical system. The Thomas-Fermi approximation can be justified by reducing the existence problem to the Painlev\'e-II equation, which admits a unique global Hastings-McLeod solution. We illustrate numerically that an iterative approach to solving the existence problem converges but give no analytical proof of this result. Generalizations are discussed for the stationary Gross-Pitaevskii equation with harmonic real and localized imaginary potentials.