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The ground state of the energy super-critical Gross--Pitaevskii equation witha harmonic potential converges in the energy space to the singular solution inthe limit of large amplitudes. The ground state can be represented by asolution curve which has either oscillatory or monotone behavior, depending onthe dimension of the system and the power of the focusing nonlinearity. Weaddress here the monotone case for the cubic nonlinearity in the spatialdimensions $d \geq 13$. By using the shooting method for the radialSchr\"{o}dinger operators, we prove that the Morse index of the ground state isfinite and is independent of the (large) amplitude. It is equal to the Morseindex of the limiting singular solution, which can be computed from numericalapproximations. The numerical results suggest that the Morse index of theground state is one and that it is stable in the time evolution of the cubicGross--Pitaevskii equation in dimensions $d \geq 13$.

Morse index for the ground state in the energy supercritical Gross--Pitaevskii equation