Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential

Burq, Nicolas; Planchon, Fabrice; Stalker, John; Tahvildar-Zadeh, A. Shadi

doi:10.1016/s0022-1236(03)00238-6

Cited by 244 publications

(337 citation statements)

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“…Several results are available for the equations i∂ t u − ∆u + V (x)u = 0, u + V (x)u = 0. We cite among the others [8], [15], [16], [19], [32] and the recent survey [33] for Schrö-dinger; and [5], [6], [10], [12], [13] for the wave equation. We must also mention the wave operator approach of Yajima (see [2], [39], [40], [41]) which permits to deal with the above equations in a unified way, although under nonoptimal assumptions on the potential in dimensions 1 and 3.…”

Section: Introductionmentioning

confidence: 99%

Decay estimates for the wave and Dirac equations with a magnetic potential

D’Ancona

Fanelli

2006

Comm Pure Appl Math

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Section: Introductionmentioning

confidence: 99%

Decay estimates for the wave and Dirac equations with a magnetic potential

D’Ancona

Fanelli

2006

Comm Pure Appl Math

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“…We also notice that a similar type of Strichartz estimates are treated in [19]. See also [3]. This paper is organized as follows.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A generalization of the weighted Strichartz estimates for wave equations and an application to self‐similar solutions

Kato

Nakamura

Ozawa

2006

Comm Pure Appl Math

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Weighted Strichartz estimates with homogeneous weights with critical exponents are proved for the wave equation without support restriction on the forcing term. The method of proof is based on the expansion by spherical harmonics and on the Sobolev space over the unit sphere, by which the required estimates are reduced to the radial case. As an application of the weighted Strichartz estimates, the existence and uniqueness of self-similar solutions to nonlinear wave equations is proved up to 5 space dimensions.

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“…The result remains true for µ = 2 ( [5,6]), but in [32], the authors prove that for repulsive potentials which are homogeneous of degree smaller than 2, global Strichartz estimates fail to exist.…”

Section: Corollary 22 Under the Assumptions Of Proposition 21 Zermentioning

confidence: 96%

On semi-classical limit of nonlinear quantum scattering

Carles¹

2016

Ann. Sci. École Norm. Sup.

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ABSTRACT. We consider the nonlinear Schrödinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a complete scattering theory is available, showing that both the potential and the nonlinearity are asymptotically negligible for large time. Then, for data under the form of coherent state, we show that a scattering theory is also available for the approximate envelope of the propagated coherent state, which is given by a nonlinear equation. In the semi-classical limit, these two scattering operators can be compared in terms of classical scattering theory, thanks to a uniform in time error estimate. Finally, we infer a large time decoupling phenomenon in the case of finitely many initial coherent states.

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Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential

Abstract: We prove spacetime weighted-L 2 estimates for the Schrödinger and wave equation with an inverse-square potential. We then deduce Strichartz estimates for these equations.

Cited by 244 publications

References 32 publications

Decay estimates for the wave and Dirac equations with a magnetic potential

Decay estimates for the wave and Dirac equations with a magnetic potential

A generalization of the weighted Strichartz estimates for wave equations and an application to self‐similar solutions

On semi-classical limit of nonlinear quantum scattering

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