Remark on the analytic smoothing for the Schrödinger equation (Q2733840)

Let \(g_0\) be the flat metric on \(\mathbb{R}^n\) and \(g\) an analytic Riemannian metric such that \(g\geq\nu g_0\) for some \(\nu>0\). Let \(\Delta_g\) the corresponding Laplacian. The authors consider here the operator with analytic coefficients NEWLINE\[NEWLINEP(y,D_y)= -\Delta_g+ \sum_{|\beta|\leq 1}a_\beta(y) D_y^\beta. \tag{1}NEWLINE\]NEWLINE Let \(\rho_0= (y_0,\eta_0) \in T^*\mathbb{R}^n \setminus \{0\}\) and \(\gamma^+_{\rho_0}\) be the forward bicharacteristic of \(P\) starting from \(\rho_0\). Then they consider NEWLINE\[NEWLINE\begin{cases} {\partial u\over\partial t}+iP(y,D_y) u=0, \\ u|_{t=0}= u_0\end{cases} \tag{2}NEWLINE\]NEWLINE with \(u_0\in L^2(\mathbb{R}^n)\) and prove that a solution under some natural assumption on the data (1) and (2) possess microlocal analytic smoothness.