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 $B\in C^\infty_{\rm{pol}}(\mathbb R^d)$ and an electric potential $V\in L^1_{\rm{loc}}(\mathbb R^d)$, $V_-\in L^d(\mathbb R^d)\cap L^{d/2}(\mathbb R^d)$. Compared to the nonrelativistic case, this estimation involves both norms of $V_-$ in $L^{d/2}(\mathbb R^d)$ and in $L^{d}(\mathbb R^d)$. A direct consequence is a Lieb-Thirring inequality for the sum of powers of the absolute values of the negative eigenvalues.