Analytic wave front set for solutions to Schrödinger equations (Q734904)

The authors consider short-range type perturbations \(H\) of the Laplacian on \(\mathbb{R}^n\) and characterize the analytic wave front set of the solution to the Schrödinger equation \(e^{-itH}f\), in terms of that of the ree solution \(e^{-itH_0}\) for \(t< 0\) \((t> 0)\) in the forward (backward) non-trapping region. The result is an analytic analogue of a result by the second author [J. Funct. Anal. 256, No. 4, 1299--1309 (2009; Zbl 1155.35014)]. To be more specific, the operator \[ H= {1\over 2}\sum_{1\leq j,k\leq n}D_j a_{jk}(a)D_k+{1\over 2} \sum_{1\leq j\leq n} (a_j(n)D_j+ D_j a_j(x))+ a_0(x) \] is a short perturbation of \(H_0:= -{1\over 2}\Delta\), the coefficients \(a_\alpha\in{\mathcal C}^\infty(\mathbb{R}^n)\), are real-valued and can be extended to a holomorphic function on \(\Gamma_\nu:= \{z\in\mathbb{C}^n;|\text{Im\,}z|< \nu\langle\text{Re\,}z\rangle\}\) with some \(\nu> 0\). Let \(p(x,\xi)\) be the principal symbol of \(H\) and \((y(t,x,\xi), \eta(t,x,\xi)):= \exp(tH_p(x,\xi))\) the solution of the Hamilton system defined by \(p\), with initial data \((x,\xi)\). Let \((x_0,\xi_0)\in T^* \mathbb{R}^n\setminus 0\) be a forward non-trapping point (that is \(|y(t,x,\xi)|\to \infty\) as \(t\to\infty\)). In this case there exist \(x_+(x_0,\xi_0)\), \(\xi_+(x_0,\xi)\in \mathbb{R}^n\) such that \[ |x_+(x_0, \xi_0)+ t\xi_x(x_0, \xi_0)- y(t,x_0,\xi_0)|\to 0\quad\text{as }t\to+\infty. \] The main result is the following: for any \(t< 0\) and any \(x_0\in L^2(\mathbb{R}^d)\), one has the equivalence \[ (x_0,\xi_0)\in WF_a(e^{-itH}u_0)\Leftrightarrow (x_+(x_0, \xi_0), \xi_+(x_0, \xi_0))\in WF_a(e^{-itH_0} u_0). \]