The form boundedness criterion for the relativistic Schrödinger operator

Vladimir Maz'ya[1]; Igor Verbitsky

  • [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 (2004)

  • Volume: 54, Issue: 2, page 317-339
  • ISSN: 0373-0956

Abstract

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We establish necessary and sufficient conditions on the real- or complex-valued potential Q defined on ℝ n for the relativistic Schrödinger operator - 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 ) .

How to cite

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Maz'ya, Vladimir, and Verbitsky, Igor. "The form boundedness criterion for the relativistic Schrödinger operator." Annales de l’institut Fourier 54.2 (2004): 317-339. <http://eudml.org/doc/116113>.

@article{Mazya2004,
abstract = {We establish necessary and sufficient conditions on the real- or complex-valued potential $Q$ defined on $\{\mathbb \{R\}\}^n$ for the relativistic Schrödinger operator $\sqrt\{-\Delta \} + Q$ to be bounded as an operator from the Sobolev space $W^\{1/2\}_2 (\{\mathbb \{R\}\}^n)$ to its dual $W^\{-1/2\}_2 (\{\mathbb \{R\}\}^n)$.},
affiliation = {Linköping University, Department of Mathematics, Linköping 581-83 (Suède), University of Missouri, Department of Mathematics, Columbia, MO 65211 (USA)},
author = {Maz'ya, Vladimir, Verbitsky, Igor},
journal = {Annales de l’institut Fourier},
keywords = {relativistic Schrödinger operator; complex-valued potentials; Sobolev spaces},
language = {eng},
number = {2},
pages = {317-339},
publisher = {Association des Annales de l'Institut Fourier},
title = {The form boundedness criterion for the relativistic Schrödinger operator},
url = {http://eudml.org/doc/116113},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Maz'ya, Vladimir
AU - Verbitsky, Igor
TI - The form boundedness criterion for the relativistic Schrödinger operator
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 2
SP - 317
EP - 339
AB - We establish necessary and sufficient conditions on the real- or complex-valued potential $Q$ defined on ${\mathbb {R}}^n$ for the relativistic Schrödinger operator $\sqrt{-\Delta } + Q$ to be bounded as an operator from the Sobolev space $W^{1/2}_2 ({\mathbb {R}}^n)$ to its dual $W^{-1/2}_2 ({\mathbb {R}}^n)$.
LA - eng
KW - relativistic Schrödinger operator; complex-valued potentials; Sobolev spaces
UR - http://eudml.org/doc/116113
ER -

References

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  1. M. Aizenman, B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math 35 (1982), 209-273 Zbl0459.60069MR644024
  2. S.-Y. A. Chang, J.M. Wilson, T.H. Wolff, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv 60 (1985), 217-246 Zbl0575.42025MR800004
  3. M. Combescure, J. Ginibre, Spectral and scattering theory for the Schrödinger operator with strongly oscillating potentials, Ann. Inst. Henri Poincaré, Sec. A, Physique théorique 24 (1976), 17-29 Zbl0336.47007MR400976
  4. D.E. Edmunds, W.D. Evans, Spectral Theory and Differential Operators, (1987), Clarendon Press, Oxford Zbl0628.47017MR929030
  5. C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc 9 (1983), 129-206 Zbl0526.35080MR707957
  6. R. Kerman, E. Sawyer, The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier, Grenoble 36 (1987), 207-228 Zbl0591.47037MR867921
  7. D.S. Kurtz, R.L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc 255 (1979), 343-362 Zbl0427.42004MR542885
  8. E.H. Lieb, M. Loss, Analysis, (2001), Amer. Math. Soc., Providence, RI Zbl0873.26002
  9. V.G. Maz'ya, On the theory of the n -dimensional Schrödinger operator, Izv. Akad. Nauk SSSR, ser. Matem. 28 (1964), 1145-1172 Zbl0148.35602MR174879
  10. V.G. Maz'ya, Sobolev Spaces, (1985), Springer-Verlag, Berlin--Heidelberg--New York Zbl0692.46023MR817985
  11. V.G. Maz'ya, T.O. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, 23 (1985), Pitman, Boston--London Zbl0645.46031MR881055
  12. V.G. Maz'ya, I.E. Verbitsky, Capacitary estimates for fractional integrals, with applications to partial differential equations and Sobolev multipliers, Arkiv för Matem 33 (1995), 81-115 Zbl0834.31006MR1340271
  13. V.G. Maz'ya, I.E. Verbitsky, The Schrödinger operator on the energy space: boundedness and compactness criteria, Acta Math 188 (2002), 263-302 Zbl1013.35021MR1947894
  14. V.G. Maz'ya, I.E. Verbitsky, Boundedness and compactness criteria for the one-dimensional Schrödinger operator, Function Spaces, Interpolation Theory and Related Topics (2002), 369-382, De Gruyter, Berlin Zbl1034.34101MR1943294
  15. E. Nelson, Topics in Dynamics. I: Flows, (1969), Princeton University Press, Princeton, New Jersey Zbl0197.10702MR282379
  16. M. Reed, B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, (1975), Academic Press, New York--London Zbl0308.47002MR493420
  17. M. Schechter, Operator Methods in Quantum Mechanics, (1981), North-Holland, Amsterdam -- New York -- Oxford Zbl0456.47012MR597895
  18. B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc 7 (1982), 447-526 Zbl0524.35002MR670130
  19. E.M. Stein, Singular Integrals and Differentiability Properties of Functions, (1970), Princeton University Press, Princeton, New Jersey Zbl0207.13501MR290095
  20. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, (1993), Princeton University Press, Princeton, New Jersey Zbl0821.42001MR1232192
  21. I.E. Verbitsky, Nonlinear potentials and trace inequalities, Operator Theory: Advances and Applications 110 (1999), 323-343 Zbl0941.31001MR1747901

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