Gabor analysis for Schrodinger equations and propagation of singularities

We consider the Schrödinger equation \begin{equation*} i \displaystyle\frac{\partial u}{\partial t} +Hu=0,\quad H=a(x,D), \end{equation*} where the Hamiltonian $a(z)$, $z=(x,ξ)$, is assumed real-valued and smooth, with bounded derivatives $|\partial^αa(z)|\leq C_α$, for every $|α|\geq 2$, $z\in\mathbb{R}^{2d}$. For such equation results are known concerning well-posedness of the Cauchy problem for initial data in $L^2(\mathbb{R}^d)$ and local representation of the propagator $e^{it H}$ by means of Fourier integral operators. In the present paper we give a global expression for $e^{itH}$ in terms of Gabor analysis and we deduce boundedness in modulation spaces. Moreover, by using time-frequency techniques, we obtain a result of propagation of micro-singularities for $e^{itH}$.

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