The form boundedness criterion for the relativistic Schrödinger operator

Maz'ya, Vladimir; Verbitsky, Igor

doi:10.5802/aif.2020

The form boundedness criterion for the relativistic Schrödinger operator
[Un critère de bornitude pour l'opérateur de Schrödinger relativiste]

Maz'ya, Vladimir ¹ ; Verbitsky, Igor

¹ Linköping University, Department of Mathematics, Linköping 581-83 (Suède), University of Missouri, Department of Mathematics, Columbia, MO 65211 (USA)

Annales de l'Institut Fourier, Tome 54 (2004) no. 2, pp. 317-339.

Résumé
Abstract

Nous donnons des conditions nécessaires et suffisantes sur le potentiel $Q$ , défini sur $ℝ^{n}$ et à valeurs réelles ou complexes, pour que l’opérateur de Schrödinger relativiste $\sqrt{- Δ} + Q$ soit un opérateur borné de l’espace de Sobolev $W_{2}^{1 / 2} (ℝ^{n})$ dans son dual $W_{2}^{- 1 / 2} (ℝ^{n})$ .

We establish necessary and sufficient conditions on the real- or complex-valued potential $Q$ defined on $ℝ^{n}$ for the relativistic Schrödinger operator $\sqrt{- Δ} + Q$ to be bounded as an operator from the Sobolev space $W_{2}^{1 / 2} (ℝ^{n})$ to its dual $W_{2}^{- 1 / 2} (ℝ^{n})$ .

Zbl

DOI : 10.5802/aif.2020

Classification : 35J10, 31C15, 42B15, 46E35
Keywords: relativistic Schrödinger operator, complex-valued potentials, Sobolev spaces
Mot clés : opérateur de Schrödinger relativiste, potentiels à valeurs complexes, espaces de Sobolev

Affiliations des auteurs :

Maz'ya, Vladimir ¹ ; Verbitsky, Igor

¹ Linköping University, Department of Mathematics, Linköping 581-83 (Suède), University of Missouri, Department of Mathematics, Columbia, MO 65211 (USA)

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     author = {Maz'ya, Vladimir and Verbitsky, Igor},
     title = {The form boundedness criterion for the relativistic {Schr\"odinger} operator},
     journal = {Annales de l'Institut Fourier},
     pages = {317--339},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {2},
     year = {2004},
     doi = {10.5802/aif.2020},
     zbl = {02123569},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.2020/}
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Maz'ya, Vladimir; Verbitsky, Igor. The form boundedness criterion for the relativistic Schrödinger operator. Annales de l'Institut Fourier, Tome 54 (2004) no. 2, pp. 317-339. doi : 10.5802/aif.2020. https://www.numdam.org/articles/10.5802/aif.2020/

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