Sparsity of Gabor representation of Schrodinger propagators

Recent papers show how tight frames of curvelets and shearlets provide optimally sparse representation of hyperbolic-type Fourier integral operators (FIOs) [E.J. Candes, L. Demanet, Curvelets and Fourier integral operators, C. R. Math. Acad. Sci. Paris 336 (5) (2003) 395-398; E.J. Candes, L. Demanet, The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math. 58 (2005) 1472-1528; E.J. Candes, 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, Theorie spectrale pour des operateurs globalement elliptiques, Asterisque, Societe Mathematique de France. 1984: B. Helffer, D. Robert, Comportement asymptotique precise du spectre d'operateurs globalement elliptiques dans R(d), Sem. Goulaouic-Meyer-Schwartz 1980-81, Ecole Polytechnique, 1980, Expose 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 Schrodinger equation with a quadratic Hamiltonian. We show that Gabor frames provide optimally sparse representations of such operators. Numerical examples for the Schrodinger case demonstrate the fast computation of these operators. (C) 2008 Elsevier Inc. All rights reserved.