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We prove that the non-radial sign-changing solutions to the nonlinearSchr\"odinger equation \begin{equation*} \Delta u-u+|u|^{p-1}u=0 \mbox{ in}\R^N, \quad u \in H^1 (\R^N ) \end{equation*} constructed by Musso, Pacard andWei is non-degenerate. This provides the first example of non-degeneratesign-changing solution with finite energy to the above nonlinear Schr\"odingerequation.

Nondegeneracy of nonradial sign-changing solutions to the nonlinear Schr\"odinger equations