Propagation of global analytic singularities for Schrödinger equations with quadratic Hamiltonians

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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 the initial value problem { u t + q w ( x , D ) u = 0 u | t = 0 = u 0 , on R n , where u 0 is a tempered distribution on R n , q = q ( x , ξ ) is a complex-valued quadratic form on R 2 n = R x n × R ξ n with nonnegative real part Re q ≥ 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 R 2 n , called the singular space of q , are transported by the Hamilton flow of 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.