Unimodular Fourier multipliers for modulation spaces

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol $e^{i|\xi|^\alpha}$, where $\alpha\in[0, 2]$, are bounded on all modulation spaces, but, in general, fail to be bounded on the usual $L^p$-spaces. As a consequence, the phase-space concentration of the solutions to the free Schr\"odinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers $|\xi|^{-\delta} \sin (|\xi|^\alpha)$ for $0 \leq \delta \leq \alpha$.