Propagation of global analytic singularities for Schrodinger equations with quadratic Hamiltonians

摘要

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 u0 is a tempered distribution on R-n, q = q(x, xi) is a complex-valued quadratic formon R-2n = R-x(n) x R-xi(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-2n, 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. (C) 2022 The Author(s). Published by Elsevier Inc.