Sharp non-uniqueness of singular solutions to the one-dimensional periodic cubic nonlinear Schrödinger equation

We construct by convex integration a nonzero singular weak solution of the one-dimensional periodic cubic nonlinear Schrödinger equation that is compactly supported in time, has zero initial datum, and satisfies \[ u\in\bigcap_{α<1/6} C_t^0 H_x^α\cap \bigcap_{1\leq p<3} C_t^0 L_x^p. \] The key new i