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 $e^{tq^{\mathrm{w}}}$ if $\mathrm{Re} (q) \le 0$ and the companion singular space associated is trivial. This result is then leveraged to achieve a comprehensive analysis of the phase regularity of $e^{tq^{\mathrm{w}}}$ with $\mathrm{Re} (q) \le 0$, thereby extending the $L^2$ analysis of quadratic semigroups initiated by Hitrik and Pravda-Starov to general modulation spaces $M^p(\mathbb{R}^d)$, $1 \le p \le \infty$, with optimal explicit bounds.