Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian

The Berezin–Li–Yau and the Kröger inequalities show that Riesz means of order $\ge 1$ of the eigenvalues of the Laplacian on a domain $\mathrm{\Omega }$ of finite measure are bounded in terms of their semiclassical limit expressions. We show that these inequalities can be improved by a multiplicative factor that depends only on the dimension and the product $\sqrt{\mathrm{\Lambda }}\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}d}$, where $\mathrm{\Lambda }$ is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when $\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}d}$ is replaced by a generalized inradius of $\mathrm{\Omega }$. Finally, we show similar inequalities in two dimensions in the presence of a constant magnetic field.