About a non-homogeneous Hardy-inequality and its relation with the spectrum of Dirac operators

Jean Dolbeault[1]; Maria J. Esteban[1]; Eric Séré[1]

  • [1] CEREMADE (UMR CNRS 7534) Université Paris-Dauphine F-75775 Paris Cedex 16

Séminaire Équations aux dérivées partielles (2001-2002)

  • Volume: 2001-2002, page 1-10

Abstract

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A non-homogeneous Hardy-like inequality has recently been found to be closely related to the knowledge of the lowest eigenvalue of a large class of Dirac operators in the gap of their continuous spectrum.

How to cite

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Dolbeault, Jean, Esteban, Maria J., and Séré, Eric. "About a non-homogeneous Hardy-inequality and its relation with the spectrum of Dirac operators." Séminaire Équations aux dérivées partielles 2001-2002 (2001-2002): 1-10. <http://eudml.org/doc/11037>.

@article{Dolbeault2001-2002,
abstract = {A non-homogeneous Hardy-like inequality has recently been found to be closely related to the knowledge of the lowest eigenvalue of a large class of Dirac operators in the gap of their continuous spectrum.},
affiliation = {CEREMADE (UMR CNRS 7534) Université Paris-Dauphine F-75775 Paris Cedex 16; CEREMADE (UMR CNRS 7534) Université Paris-Dauphine F-75775 Paris Cedex 16; CEREMADE (UMR CNRS 7534) Université Paris-Dauphine F-75775 Paris Cedex 16},
author = {Dolbeault, Jean, Esteban, Maria J., Séré, Eric},
journal = {Séminaire Équations aux dérivées partielles},
language = {eng},
pages = {1-10},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {About a non-homogeneous Hardy-inequality and its relation with the spectrum of Dirac operators},
url = {http://eudml.org/doc/11037},
volume = {2001-2002},
year = {2001-2002},
}

TY - JOUR
AU - Dolbeault, Jean
AU - Esteban, Maria J.
AU - Séré, Eric
TI - About a non-homogeneous Hardy-inequality and its relation with the spectrum of Dirac operators
JO - Séminaire Équations aux dérivées partielles
PY - 2001-2002
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2001-2002
SP - 1
EP - 10
AB - A non-homogeneous Hardy-like inequality has recently been found to be closely related to the knowledge of the lowest eigenvalue of a large class of Dirac operators in the gap of their continuous spectrum.
LA - eng
UR - http://eudml.org/doc/11037
ER -

References

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  2. S.N. Datta and G. Deviah. The minimax technique in relativistic Hartree-Fock calculations. Pramana, 30(5) (1988), p.387-405. 
  3. J. Dolbeault, M.J. Esteban and E. Séré. Variational characterization for eigenvalues of Dirac operators. Cal. Var. 10 (2000), p. 321-347. Zbl0968.49025MR1767717
  4. J. Dolbeault, M.J. Esteban, E. Séré. On the eigenvalues of operators with gaps. Application to Dirac operators. J. Funct. Anal. 174 (2000), p. 208-226. Zbl0982.47006MR1761368
  5. J. Dolbeault, M.J. Esteban, E. Séré, M. Vanbreugel. Minimization methods for the one-particle Dirac equation. Phys. Rev. Letters 85(19) (2000), p. 4020-4023. 
  6. J. Dolbeault, M.J. Esteban, E. Séré. A variational method for relativistic computations in atomic and molecular physics. To appear in Int. J. Quantum Chemistry. 
  7. M.J. Esteban, E. Séré. Existence and multiplicity of solutions for linear and nonlinear Dirac problems. Partial Differential Equations and Their Applications. CRM Proceedings and Lecture Notes, volume 12. Eds. P.C. Greiner, V. Ivrii, L.A. Seco and C. Sulem. AMS, 1997. Zbl0889.35084MR1479240
  8. M. Griesemer, R.T. Lewis, H. Siedentop. A minimax principle in spectral gaps: Dirac operators with Coulomb potentials. Doc. Math. 4 (1999), P. 275-283 (electronic). Zbl1115.47300MR1701887
  9. M. Griesemer, H. Siedentop. A minimax principle for the eigenvalues in spectral gaps. J. London Math. Soc. (2) 60 no. 2 (1999), p. 490-500. Zbl0952.47022MR1724845
  10. M. Klaus and R. Wüst. Characterization and uniqueness of distinguished self-adjoint extensions of Dirac operators. Comm. Math. Phys. 64(2) (1978-79), p. 171-176. Zbl0408.47022MR519923
  11. G. Nenciu. Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms. Comm. Math. Phys. 48 (1976), p. 235-247. Zbl0349.47014MR421456
  12. U.W. Schmincke. Distinguished self-adjoint extensions of Dirac operators. Math. Z. 129 (1972), p. 335-349. Zbl0252.35062MR326448
  13. J.D. Talman. Minimax principle for the Dirac equation. Phys. Rev. Lett. 57(9) (1986), p. 1091-1094. MR854208
  14. B. Thaller. The Dirac Equation. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1st edition, 1992. Zbl0765.47023MR1219537
  15. C. Tix. Strict Positivity of a relativistic Hamiltonian due to Brown and Ravenhall. Bull. London Math. Soc. 30(3) (1998), p. 283-290. Zbl0939.35134MR1608118
  16. R. Wüst. Dirac operators with strongly singular potentials. M ath. Z. 152 (1977), p. 259-271. Zbl0361.35051MR437948

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