On the Thomas-Fermi Approximation of the Ground State in a PT-Symmetric Confining Potential
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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.
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