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 $B\in C_{\mathrm{pol}}^{\infty }({\mathbb{R}}^d)$ and an electric potential $V\in L_{\mathrm{loc}}^1({\mathbb{R}}^d)$, $V_{-}\in L^d({\mathbb{R}}^d)\cap L^{d/2}({\mathbb{R}}^d)$. 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.