Dispersive smoothing for Schrödinger equations (Q1919329)

The authors investigate conditions on the positive symmetric matrix \((a_{i,j} (x))\) that guarantee infinite differentiability of the fundamental solution of the Schrödinger equation \[ \left( {1\over i} {\partial\over\partial t} - \sum^d_{i,j=1} {\partial \over\partial x_i} a_{ij} (x) {\partial \over\partial x_j} \right) \psi(t,x) = 0, \quad x \in\mathbb{R}^d,\psi (0,x) =f(x) \] with respect to \(t\) and \(x\), when \(t>0\) and \(x\in \mathbb{R}^d\).