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

Abstract

We prove the analog of the Cwickel-Lieb-Rosenblum estimation for the number of negative eigenvalues of a relativistic Hamiltonian with magnetic field B 2 C1 pol(Rd) and an electric potential V 2 L1 loc(Rd), V− 2 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.