The form boundedness criterion for the relativistic Schrödinger operator

Maz'ya, Vladimir; Verbitsky, Igor

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

Maz'ya, Vladimir ; Verbitsky, Igor

Annales de l'Institut Fourier, Tome 54 (2004), p. 317-339 / Harvested from Numdam

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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 $W_{2}^{- 1 / 2} (ℝ^{n})$ .

Publié le : 2004-01-01

Zbl 02123569

DOI : https://doi.org/10.5802/aif.2020
Classification: 35J10, 31C15, 42B15, 46E35
Mots clés: opérateur de Schrödinger relativiste, potentiels à valeurs complexes, espaces de Sobolev

@article{AIF_2004__54_2_317_0,
     author = {Maz'ya, Vladimir and Verbitsky, Igor},
     title = {The form boundedness criterion for the relativistic Schr\"odinger operator},
     journal = {Annales de l'Institut Fourier},
     volume = {54},
     year = {2004},
     pages = {317-339},
     doi = {10.5802/aif.2020},
     zbl = {02123569},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2004__54_2_317_0}
}

Maz'ya, Vladimir; Verbitsky, Igor. The form boundedness criterion for the relativistic Schrödinger operator. Annales de l'Institut Fourier, Tome 54 (2004) pp. 317-339. doi : 10.5802/aif.2020. http://gdmltest.u-ga.fr/item/AIF_2004__54_2_317_0/

Bibliographie

[AiS] M. Aizenman; B. Simon Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math, Tome 35 (1982), pp. 209-273 | MR 644024 | Zbl 0459.60069

[ChWW] S.-Y. A. Chang; J.M. Wilson; T.H. Wolff Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv, Tome 60 (1985), pp. 217-246 | MR 800004 | Zbl 0575.42025

[CoG] M. Combescure; J. Ginibre Spectral and scattering theory for the Schrödinger operator with strongly oscillating potentials, Ann. Inst. Henri Poincaré, Sec. A, Physique théorique, Tome 24 (1976), pp. 17-29 | Numdam | MR 400976 | Zbl 0336.47007

[EE] D.E. Edmunds; W.D. Evans Spectral Theory and Differential Operators, Clarendon Press, Oxford (1987) | MR 929030 | Zbl 0628.47017

[Fef] C. Fefferman The uncertainty principle, Bull. Amer. Math. Soc, Tome 9 (1983), pp. 129-206 | MR 707957 | Zbl 0526.35080

[KeS] R. Kerman; E. Sawyer The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier, Grenoble, Tome 36 (1987), pp. 207-228 | Numdam | MR 867921 | Zbl 0591.47037

[KWh] D.S. Kurtz; R.L. Wheeden Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc, Tome 255 (1979), pp. 343-362 | MR 542885 | Zbl 0427.42004

[LL] E.H. Lieb; M. Loss Analysis, Amer. Math. Soc., Providence, RI (2001) | Zbl 0873.26002

[M1] V.G. Maz'Ya On the theory of the $n$ -dimensional Schrödinger operator, Izv. Akad. Nauk SSSR, ser. Matem., Tome 28 (1964), pp. 1145-1172 | MR 174879 | Zbl 0148.35602

[M2] V.G. Maz'Ya Sobolev Spaces, Springer-Verlag, Berlin--Heidelberg--New York (1985) | MR 817985 | Zbl 0692.46023

[MSh] V.G. Maz'Ya; T.O. Shaposhnikova Theory of Multipliers in Spaces of Differentiable Functions, Pitman, Boston--London, Monographs and Studies in Mathematics, Tome 23 (1985) | MR 881055 | Zbl 0645.46031

[MV1] V.G. Maz'Ya; I.E. Verbitsky Capacitary estimates for fractional integrals, with applications to partial differential equations and Sobolev multipliers, Arkiv för Matem, Tome 33 (1995), pp. 81-115 | MR 1340271 | Zbl 0834.31006

[MV2] V.G. Maz'Ya; I.E. Verbitsky The Schrödinger operator on the energy space: boundedness and compactness criteria, Acta Math, Tome 188 (2002), pp. 263-302 | MR 1947894 | Zbl 1013.35021

[MV3] V.G. Maz'Ya; I.E. Verbitsky; Eds. M. Cwikel, A. Kufner, G. Sparr. Boundedness and compactness criteria for the one-dimensional Schrödinger operator, Function Spaces, Interpolation Theory and Related Topics, De Gruyter, Berlin (Proc. Jaak Peetre Conf. (Lund, Sweden, August 17-22, 2000)) (2002), pp. 369-382 | MR 1943294 | Zbl 1034.34101

[Nel] E. Nelson Topics in Dynamics. I: Flows, Princeton University Press, Princeton, New Jersey (1969) | MR 282379 | Zbl 0197.10702

[RS] M. Reed; B. Simon Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, Academic Press, New York--London (1975) | MR 493420 | Zbl 0308.47002

[Sch] M. Schechter Operator Methods in Quantum Mechanics, North-Holland, Amsterdam -- New York -- Oxford (1981) | MR 597895 | Zbl 0456.47012

[Sim] B. Simon Schrödinger semigroups, Bull. Amer. Math. Soc, Tome 7 (1982), pp. 447-526 | MR 670130 | Zbl 0524.35002

[St1] E.M. Stein Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey (1970) | MR 290095 | Zbl 0207.13501

[St2] E.M. Stein Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey (1993) | MR 1232192 | Zbl 0821.42001

[Ver] I.E. Verbitsky; Eds. J. Rossmann, P. Takác And G. Wildenhain Nonlinear potentials and trace inequalities, Operator Theory: Advances and Applications (The Maz'ya Anniversary Collection) Tome 110 (1999), pp. 323-343 | MR 1747901 | Zbl 0941.31001