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May  2008, 7(3): 533-562. doi: 10.3934/cpaa.2008.7.533

Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators

Roberta Bosi 1, , Jean Dolbeault 2, and  Maria J. Esteban 3,

1. 

Institut für Analysis und Scientific Computing, Wiedner Hauptstr. 8, A 1040 Wien, Österreich, Austria
2. 

Ceremade (UMR CNRS no. 7534), Université Paris Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16
3. 

CEREMADE - UMR C.N.R.S. 7534, Université Paris IX-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

Received  November 2006 Revised  August 2007 Published  February 2008

By expanding squares, we prove several Hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities. These inequalities involve the $L^2$ norm. Such results are generalized to an arbitrary number of singularities and compared with standard results given by the IMS method. The generalized version of Hardy inequalities with several singularities is equivalent to some spectral information on a Schrödinger operator involving a potential with several inverse square singularities. We also give a generalized Hardy inequality for Dirac operators in the case of a potential having several singularities of Coulomb type, which are critical for Dirac operators.
Keywords: Hardy inequalities, optimal inequalities, Schrödinger operator, Dirac-Coulomb Hamiltonian., singular potentials, weighted norms.
Mathematics Subject Classification: Primary: 26D10, 35J10, 46E35; Secondary: 35P15, 35Q40, 46N50, 47N50, 81V45, 81V5.
Citation: Roberta Bosi, Jean Dolbeault, Maria J. Esteban. Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators. Communications on Pure & Applied Analysis, 2008, 7 (3) : 533-562. doi: 10.3934/cpaa.2008.7.533
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