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 $\Omega$ 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 Λ∣Ω∣1/d, where $\Lambda$ is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when ∣Ω∣1/d is replaced by a generalized inradius of $\Omega$. Finally, we show similar inequalities in two dimensions in the presence of a constant magnetic field.