On the spectrum of the Dirichlet Laplacian for horn-shaped regions in Rn with infinite volume

An inequality for trace (e^tΔD) is proven, where ?Δ_D is the Dirichlet Laplacian for horn-shaped regions D in Rⁿ. The results of Rozenbljum and Simon for the leading asymptotics for the growth of the number of eigenvalues of the two-dimensional Dirichlet Laplacian in the regions {$\text{(x, y):¦x¦}{}^{\mathrm{\mu }}\text{· ¦y¦ ? 1, μ > 0}$} are easily recovered. An example of a horn-shaped region in R² where that asymptotics is exponential is given.