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For the stationary nonlinear Schr\"odinger equation $-\Delta u+ V(x)u- f(u) =\lambda u$ with periodic potential $V$ we study the existence and stabilityproperties of multibump solutions with prescribed $L^2$-norm. To this end weintroduce a new nondegeneracy condition and develop new superpositiontechniques which allow to match the $L^2$-constraint. In this way we obtain theexistence of infinitely many geometrically distinct solutions to the stationaryproblem. We then calculate the Morse index of these solutions with respect tothe restriction of the underlying energy functional to the associated$L^2$-sphere, and we show their orbital instability with respect to theSchr\"odinger flow. Our results apply in both, the mass-subcritical and themass-supercritical regime.

Unstable normalized standing waves for the space periodic NLS