Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators
Esteban, Maria J.; Dolbeault, Jean; Bosi, Roberta (2008), Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators, Communications on Pure and Applied Mathematics, 7, 3, p. 533–562
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00113158/en/
Journal nameCommunications on Pure and Applied Mathematics
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Abstract (EN)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 L2 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.
Subjects / KeywordsDirac-Coulomb Hamiltonian; singular potentials; Schrödinger operator; optimal inequalities; weighted norms; Hardy inequalities
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