Estimating the number of negative eigenvalues of a relativistic Hamiltonian with regular magnetic field

الأصل · EN

We prove the analog of the Cwickel-Lieb-Rosenblum estimation for the number of negative eigenvalues of a relativistic Hamiltonian with magnetic field B∈ C∞pol(Rᵈ) and an electric potential V∈ L¹loc(Rᵈ), V-∈ Lᵈ(Rᵈ)∩ Lᵈ/²(Rᵈ). Compared to the nonrelativistic case, this estimation involves both norms of V- in Lᵈ/²(Rᵈ) and in Lᵈ(Rᵈ). A direct consequence is a Lieb-Thirring inequality for the sum of powers of the absolute values of the negative eigenvalues.