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

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.