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 by using the unstable manifold theorem for a planar dynamical system. To justify the Thomas-Fermi approximation, the existence problem can be reduced to the Painleve-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.