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We study the propagation in time of $1/2$-Gelfand-Shilov singularities, i.e.global analytic singularities, of tempered distributional solutions of theinitial value problem \begin{align*} \begin{cases} u_t + q^w(x,D) u = 0 \\u|_{t=0} = u_0, \end{cases} \end{align*} on $\mathbb{R}^n$, where $u_0$ is atempered distribution on $\mathbb{R}^n$, $q=q(x,\xi)$ is a complex-valuedquadratic form on $\mathbb{R}^{2n} = \mathbb{R}^n_x \times \mathbb{R}^n_\xi$with nonnegative real part $\textrm{Re} \ q \ge 0$, and $q^w(x,D)$ is the Weylquantization of $q$. We prove that the $1/2$-Gelfand-Shilov singularities ofthe initial data that are contained within a distinguished linear subspace ofthe phase space $\mathbb{R}^{2n}$, called the singular space of $q$, aretransported by the Hamilton flow of $\textrm{Im} \ q$, while all other$1/2$-Gelfand-Shilov singularities are instantaneously regularized. Our resultextends the observation of Hitrik, Pravda-Starov, and Viola '18 that thisevolution is instantaneously globally analytically regularizing when thesingular space of $q$ is trivial.

Propagation of Global Analytic Singularities for Schr\"odinger Equations with Quadratic Hamiltonians