Propagation of analytic singularities for short and long range perturbations of the free Schrödinger equation (Q2419138)

This paper deals with the analytic singularities of the distributions \(u(t,x)\) satisfying in \(\mathbb{R}^1\times \mathbb{R}^n\) the Schrödinger equation \[i\frac{\partial u}{\partial t}= Pu,\quad u|_{t=0}= u_0(x),\tag{1}\] where \(P(x,D_x)\) is a second-order symmetric differential operator on \(\mathbb{R}^n\) with analytic coefficients. More precisely, it is a perturbation of \(P_0=-\frac{1}{2}\Delta\) having analytic coefficients in \[\Gamma_\nu= \{z\in \mathbb{C}^n:|\text{Im\,}z|<\nu< \text{Re\,}z\},\quad \nu>0,\] \(u_0\in L^2(\mathbb{R}^n)\), and the coefficients \(a_{jk}\) of the principal symbol \(p(x,\xi)\) of \(P\) are real-valued on \(\mathbb{R}^n\) and satisfy in \(\Gamma_\nu\) the estimate \(|a_{jk}(x)- \delta_{j,k}|\le C_0\langle x\rangle^{-\sigma}\). The short range case appears for \(\sigma>1\), while \(\sigma\in(0, 1]\) is called the long range case. Let \(P= P_0\) and \(u_0=(-2i\pi)^{n/2} e^{-i|x|^2/2}\). Then the corresponding solution \(u(t)\) of (1) is such that \(u(1)= \delta(x)\), while \(u(0)\) is analytic in \(x\). The question to appear is how to express the analytic singularities of \(u(t)\) via the singularities of \(u_0\). The answer is positive under the so-called non-trapping conditions imposed on the bicharacteristics of \(p(x,\xi)\). The main results -- Theorem 2.1 (case \(\sigma>1\)) and Theorem 2.9 (case \(0<\sigma\le 1\)) -- are illustrated by the following example. Let \(P= P_0+ V(x)\), where the analytic function \(V(x)\to 0\) for \(|x|\to\infty\). Thus \(u(t)= e^{-itP}u_0\). For all \(t\in\mathbb{R}^1\) it is shown that \(WF_a(e^{itP_0}u(t))= WF_a(u_0)\) or, equivalently, \[WF_a(u(t))= WF_a(e^{-itP_0}u_0).\] Here \(WF_a(\cdot)\) stands for the analytical wave front set of the corresponding distribution. For the entire collection see [Zbl 1412.32002].