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C. Gallo and D. Pelinovsky

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Eigenvalues of a nonlinear ground state
in the Thomas-Fermi approximation

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J. Math. Anal. Appl. 355, 495–526 (2009)

**Abstract:**

We study a nonlinear ground state of the Gross-Pitaevskii equation with a
parabolic potential in the hydrodynamics limit often referred to
as the Thomas-Fermi approximation. Existence of the energy
minimizer has been known in literature for some time but it was
only recently when the Thomas-Fermi approximation was
rigorously justified. The spectrum of linearization of the
Gross-Pitaevskii equation at the ground state consists of
an unbounded sequence of positive eigenvalues. We analyze
convergence of eigenvalues in the hydrodynamics limit.
Convergence in norm of the resolvent operator is proved and
the convergence rate is estimated. We also study asymptotic and numerical
approximations of eigenfunctions and eigenvalues using Airy functions.

**Keywords:**

Gross-Pitaevskii equation, ground states, Airy functions,
existence and stability, convergence of resolvent