On the essential spectrum of the Laplacian and vague convergence of the curvature at infinity

We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval $[(n-1{)}^{2}K\mathrm{/}4,\mathrm{\infty })$ if an $n$-dimensional Riemannian manifold has an end and the average of the part of the Ricci curvature on the end which lies below a nonpositive constant $(n-1)K$ converges to zero at infinity.