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

We prove that the non-radial sign-changing solutions to the nonlinear Schr"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 and Wei is non-degenerate. This provides the first example of non-degenerate sign-changing solution with finite energy to the above nonlinear Schr"odinger equation.