Sparsity of Gabor representation of Schrödinger propagators

Recent papers show how tight frames of curvelets and shearlets provide optimally sparse representation of hyperbolic-type Fourier Integral Operators (FIOs). In this paper we address to another class of FIOs, employed by Helffer and Robert to study spectral properties of globally elliptic operators of Quantum Mechanics, and hence studied by many other authors. An example is provided by the resolvent of the Cauchy problem for the Schrödinger equation with a quadratic Hamiltonian. We show that Gabor frames provide optimally sparse representations of such operators. Numerical examples for the Schrödinger case demonstrate the fast computation of these operators.