For the 1D periodic cubic NLS, there exists a nonzero singular weak solution with zero initial data in every C_t^0 H_x^alpha below H^{1/6}, making the H^{1/6} uniqueness threshold sharp in the stated singular solution class.
Bourgain,Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations
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Sharp non-uniqueness of singular solutions to the one-dimensional periodic cubic nonlinear Schr\"odinger equation
For the 1D periodic cubic NLS, there exists a nonzero singular weak solution with zero initial data in every C_t^0 H_x^alpha below H^{1/6}, making the H^{1/6} uniqueness threshold sharp in the stated singular solution class.