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On Pointwise Convergence for Schrödinger Operator in a Convex Domain

  • Jiqiang ZhengEmail author
Article
  • 34 Downloads

Abstract

In this paper, we prove that the maximal inequality
$$\begin{aligned} \big \Vert \sup _{|t|<1}|e^{it\Delta _D}f(x,y)|\big \Vert _{L^2_{\mathrm{loc}}(\Omega )}\le C\Vert f\Vert _{H^s_D(\Omega )},\quad \forall ~f\in H^s_D(\Omega ) \end{aligned}$$
holds for any \(s>\tfrac{1}{2}\) with \(\Omega =\{(x,y)\in \mathbb {R}^2\mid x>0\}\) and \(\Delta _D=\partial _x^2+(1+x)\partial _y^2\). As a direct application, we obtain the pointwise convergence for the free Schrödinger equation \(i\partial _tu+\Delta _D u=0\) with initial data \(u(0)=f\) inside strictly convex domain.

Keywords

Schrödinger operator Pointwise convergence Airy function 

Mathematics Subject Classification

35Q55 33C10 42B25 

Notes

Acknowledgements

The author would like to express his gratitude to the anonymous referees for their invaluable comments and suggestions. The author would like to thank Fabrice Planchon for his helpful discussions and encouragement. The author was also partly supported by the ANR-16-TERC-0006-01, ANADEL.

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Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingChina
  2. 2.LJADUniversité Côte d’AzurNiceFrance

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