Local Dispersive And Strichartz Estimates For The Schrödinger Operator On The Heisenberg Group
Résumé
It was proved by H. Bahouri, P. Gérard and C.-J. Xu in [9] that the Schrödinger equation on the Heisenberg group H^d, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on H^d for the linear Schrödinger equation, by a refined study of the Schrödinger kernel S_t on H^d. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on H^d derived by B. Gaveau in [20], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results, we establish local Strichartz estimates and prove that the kernel S_t concentrates on quantized horizontal hyperplanes of H^d .
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