Ground states in Gross-Pitaevskii theory

Abstract

We study ground states in the nonlinear Schrödinger equation (NLS) with an isotropic harmonic potential, in energy-critical and energy-supercritical cases. In both cases, we prove existence of a family of ground states parametrized by their amplitude, together with the corresponding values of the spectral parameter. Moreover, we derive asymptotic behavior of the spectral parameter when the amplitude of ground states tends to infinity. We show that in the energy-supercritical case the family of ground states converges to a limiting singular solution and the spectral parameter converges to a nonzero limit, where the convergence is oscillatory for smaller dimensions, and monotone for larger dimensions. In the energy-critical case, we show that the spectral parameter converges to zero, with a specific leading-order term behavior depending on the spatial dimension.

Furthermore, we study the Morse index of the ground states in the energy-supercritical case. We show that in the case of monotone behavior of the spectral parameter, that is for large values of the dimension, the Morse index of the ground state is finite and independent of its amplitude. Moreover, we show that it asymptotically equals to the Morse index of the limiting singular solution. This result suggests how to estimate the Morse index of the ground state numerically.