Wave packet analysis of semigroups generated by quadratic differential operators

We perform a phase space analysis of evolution equations associated with the Weyl quantization $q^{\mathrm{w}}$ of a complex quadratic form $q$ on ${\mathbb{R}}^{2d}$ with non-positive real part. In particular, we obtain pointwise bounds for the matrix coefficients of the Gabor wave packet decomposition of the generated semigroup $e^{t{q}^{\mathrm{w}}}$ if $\mathrm{R}\mathrm{e}(q)\le 0$ and the companion singular space associated is trivial. This result is then leveraged to achieve a comprehensive analysis of the phase regularity of $e^{t{q}^{\mathrm{w}}}$ with $\mathrm{R}\mathrm{e}(q)\le 0$, thereby extending the $L^2$ analysis of quadratic semigroups initiated by Hitrik and Pravda-Starov to general modulation spaces $M^p({\mathbb{R}}^d)$, $1\le p\le \mathrm{\infty }$, with optimal explicit bounds.