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) [E.J. Candés, L. Demanet, Curvelets and Fourier integral operators, C. R. Math. Acad. Sci. Paris 336 (5) (2003) 395–398; E.J. Candés, L. Demanet, The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math. 58 (2005) 1472–1528; E.J. Candés, L. Demanet, L. Ying, Fast computation of Fourier integral operators, SIAM J. Sci. Comput. 29 (6) (2007) 2464–2493; K. Guo, D. Labate, Sparse shearlet representation of Fourier integral operators, Electron. Res. Announc. Math. Sci. 14 (2007) 7–19]. 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 [B. Helffer, Théorie spectrale pour des operateurs globalement elliptiques, Astérisque, Société Mathématique de France, 1984; B. Helffer, D. Robert, Comportement asymptotique precise du spectre d'operateurs globalement elliptiques dans Rd, Sem. Goulaouic–Meyer–Schwartz 1980–81, École Polytechnique, 1980, Exposé II], and hence studied by many other authors, see, e.g., [A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett. 4 (1997) 53–67; F. Concetti, J. Toft, Schatten–von Neumann properties for Fourier integral operators with non-smooth symbols I, Ark. Mat., in press]. 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.