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On the Thomas–Fermi Approximation of the Ground State in a P T ‐Symmetric Confining Potential
Author(s) -
Gallo Clément,
Pelinovsky Dmitry
Publication year - 2014
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12064
Subject(s) - space (punctuation) , ground state , fermi gamma ray space telescope , mathematics , manifold (fluid mechanics) , dimension (graph theory) , limit (mathematics) , harmonic , mathematical analysis , gross–pitaevskii equation , physics , mathematical physics , quantum mechanics , nonlinear system , pure mathematics , mechanical engineering , philosophy , linguistics , engineering
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 Painlevé‐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.