Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian

Journal article, 2026

The Berezin–Li–Yau and the Kröger inequalities show that Riesz means of order 1 of the eigenvalues of the Laplacian on a domain (formula presented) 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 (formula presented), where is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when (formula presented) is replaced by a generalized inradius of (formula presented). Finally, we show similar inequalities in two dimensions in the presence of a constant magnetic field.