Sparsity of Gabor representation of Schrödinger propagators
The authors consider Fourier integral operators given by \[ Tf(x)= \int_{\mathbb{R}^d}e^{2\pi:\Phi(x,\eta)}\sigma(x,\eta)\widehat f(\eta)d\eta,\;x\in\mathbb{R}^d, \] where the symbol \(\sigma\in BC^\infty(\mathbb{R}^{2d}\) and the phase function \(\Phi\in{\mathcal C}^\infty(\mathbb{R}^{2d};\mathbb{R})\) fulfills the estimates \(|\partial^\alpha_z\Phi(x)|\leq C_\alpha\), \(|\alpha|\geq 2\), \(z=(x,\eta)\in\mathbb{R}^{2d}\), and the nondegeneracy condition \(|\det \partial^r_{x,\eta}\Phi(x,\eta)|\geq\delta>0\). They prove the continuity of \(T\) on some modulation spaces \(M^p_\mu\), show that Gabor frames provide optimally sparse representations of such operators and present a formula which expresses \(Tf\), \(f\in M^p\equiv M^p_1\) as a finite sum of Gabor atoms, modulo an error whose size is estimated in terms of its time-frequency concentration. Numerical examples for the Schrödinger case (the free particle and the harmonic oscillator) demonstrate the fast computation of these operators.
scientific article; zbMATH DE number 5545831
scientific article