Abstract
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.
Highlights
For N ≥ 3, the simplest form of Hardy’s inequality is obtained by the “expansion of the square” method as follows: for any function u ∈ H1(RN ), 0≤ RN ∇u + α x |x|2 u dx =|∇u|2 dx + α2 − (N − 2) α |u|2 |x|2 dx, which shows for α = (N − 2)/2 that, for all u ∈ H1(RN ),|∇u|2 dx ≥ (N − 2)2, and it is well known that the constant (N − 2)2/4 is optimal
Which shows for α = (N − 2)/2 that, for all u ∈ H1(RN )
A major difference between Dirac and Schrodinger operators coupled respectively to Coulomb type interactions and to multiple inverse square singular potentials lies in the structure of the continuous spectrum
Summary
For N ≥ 3, the simplest form of Hardy’s inequality is obtained by the “expansion of the square” method as follows: for any function u ∈ H1(RN ), 0≤. Hardy inequalities; weighted norms; optimal inequalities; Schrodinger operator; singular potentials; Dirac-Coulomb Hamiltonian. A major difference between Dirac and Schrodinger operators coupled respectively to Coulomb type interactions and to multiple inverse square singular potentials lies in the structure of the continuous spectrum. This determines a minimal distance between singularities if νM > 1 Both approaches are equivalent and amount to finding estimates for the optimal constant in Hardy type inequalities for Dirac operators with multiple Coulomb singularities. Due to the non homogeneity of the Dirac operator, the best constant heavily depends on the interdistance between the singularities, and on the geometric pattern defined by them In this case we are only able to use the IMS method, which gives good results only for large interdistances, that is, for very distant singularities. The extension to the case when they differ can be worked out by the same methods, but is left to the reader
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