Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian

Journal article, 2025

The Berezin-Li-Yau and the Kroger inequalities show that Riesz means of order >= 1 of the eigenvalues of the Laplacian on a domain Q 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 root Lambda|Q|(1/d), where A is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when |Q|( 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.