Gabor analysis for Schrodinger equations and propagation of singularities

Abstract

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 $a(z)$, $z=(x,\xi )$, is assumed real-valued and smooth, with bounded derivatives $|{\partial }^{\alpha }a(z)|\le C_{\alpha }$, for every $|\alpha |\ge 2$, $z\in {\mathbb{R}}^{2d}$. 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 $e^{itH}$ 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 $e^{itH}$.

arXiv comment18 pages. Contribution for a volume dedicated to Roland Duduchava for his 70 years