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

Abstract

We study the propagation in time of $1/2$-Gelfand-Shilov singularities, i.e. global analytic singularities, of tempered distributional solutions of the initial 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 a tempered distribution on $\mathbb{R}^n$, $q=q(x,\xi)$ is a complex-valued quadratic 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 Weyl quantization of $q$. We prove that the $1/2$-Gelfand-Shilov singularities of the initial data that are contained within a distinguished linear subspace of the phase space $\mathbb{R}^{2n}$, called the singular space of $q$, are transported by the Hamilton flow of $\textrm{Im} \ q$, while all other $1/2$-Gelfand-Shilov singularities are instantaneously regularized. Our result extends the observation of Hitrik, Pravda-Starov, and Viola '18 that this evolution is instantaneously globally analytically regularizing when the singular space of $q$ is trivial.