Displaying similar documents to “Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients”

Invariant measures for the defocusing Nonlinear Schrödinger equation

Nikolay Tzvetkov (2008)

Annales de l’institut Fourier

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We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schrödinger equations on the disc of the plane ℝ 2 . We also prove an estimate giving some intuition to what may happen in 3 dimensions.

Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials

Pascal Auscher, Besma Ben Ali (2007)

Annales de l’institut Fourier

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We show various L p estimates for Schrödinger operators - Δ + V on ℝ n and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen. Our main tools are improved Fefferman-Phong inequalities and reverse Hölder estimates for weak solutions of - Δ + V and their gradients.

On the Fefferman-Phong inequality

Abdesslam Boulkhemair (2008)

Annales de l’institut Fourier

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We show that the number of derivatives of a non negative 2-order symbol needed to establish the classical Fefferman-Phong inequality is bounded by n 2 + 4 + ϵ improving thus the bound 2 n + 4 + ϵ obtained recently by N. Lerner and Y. Morimoto. In the case of symbols of type S 0 , 0 0 , we show that this number is bounded by n + 4 + ϵ ; more precisely, for a non negative symbol a , the Fefferman-Phong inequality holds if ∂ x α ∂ ξ β a ( x , ξ ) are bounded for, roughly, 4 ≤ | α | + | β | ≤ n + 4 + ϵ . To obtain such results and others, we first prove an abstract result which...

Semi-classical functional calculus on manifolds with ends and weighted L p estimates

Jean-Marc Bouclet (2011)

Annales de l’institut Fourier

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For a class of non compact Riemannian manifolds with ends, we give semi-classical expansions of bounded functions of the Laplacian. We then study related L p boundedness properties of these operators and show in particular that, although they are not bounded on L p in general, they are always bounded on suitable weighted L p spaces.

Asymptotic expansion in time of the Schrödinger group on conical manifolds

Xue Ping Wang (2006)

Annales de l’institut Fourier

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For Schrödinger operator P on Riemannian manifolds with conical end, we study the contribution of zero energy resonant states to the singularity of the resolvent of P near zero. Long-time expansion of the Schrödinger group U ( t ) = e - i t P is obtained under a non-trapping condition at high energies.