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We introduce a new notion of linear stability for standing waves of thenonlinear Schr\"odinger equation (NLS) which requires not only that thespectrum of the linearization be real, but also that the generalized kernel benot degenerate and that the signature of all the positive eigenvalues bepositive. We prove that excited states of the NLS are not linearly stable inthis more restrictive sense. We then give a partial proof that this morerestrictive notion of linear stability is a necessary condition to have orbitalstability.

On instability of excited states of the nonlinear Schr\"odinger equation