Fundamental solution of the Cauchy problem for a Schrödinger pseudo- differential operator (Q2367094)

The author considers the Schrödinger type operator \[ P(t,x,D_ t,D_ x)=D_ t+\Delta_ x+A(t,x,D_ x) \tag{1} \] where \(t\in[0,T]\), \(x\in\mathbb{R}^ n\), and \(A(t,x,D_ x)\) is a complex pseudo-differential operator defined by a symbol \(a(t,x,\xi)\) in the Hörmander class \(S^ p_{10}\), \(0<p<1\), with constants satisfying a Gevrey condition. The author constructs a fundamental solution for the Cauchy problem associated to the operator (1), with data in various spaces, among them the space of ultradistributions. To construct this fundamental solution, the author uses the so called Fourier integral operators of infinite order, as introduced in [\textit{L. Cattabriga} and \textit{L. Zanghirati}, J. Math. Kyoto Univ. 30, 149-192 (1990; Zbl 0725.35113)].