Propagation of global analytic singularities for Schrödinger 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{ut+qw(x,D)u=0u|t=0=u0, on Rn, where u0 is a tempered distribution on Rn, q=q(x,ξ) is a complex-valued quadratic form on R2n=Rxn×Rξn with nonnegative real part Req≥0, and qw(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 R2n, called the singular space of q, are transported by the Hamilton flow of Imq, 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.