The Weyl upper bound on the discrete spectrum of locally symmetric spaces.
Let \(M:=\Gamma\backslash X\) be a non-compact locally symmetric space of dimension \(n\) which has finite volume and let \(E_\sigma\) be a locally homogeneous bundle over \(M\). Let \(\Delta\) be the elliptic operator corresponding to the quadratic form \(D(f):=\int_{M}| \nabla f| ^2\) where \(f\in C_0^\infty(M,E_\sigma)\). This operator has a continuous spectrum \(\text{Spec}_{\text{con}}(\Delta)\). It also has a discrete spectrum \(\text{Spec}_{\text{dis}}(\Delta)\) of the form \(\lambda_1\leq\lambda_2\leq\dots\) Let \(N_d(\lambda):=| \{\lambda_i\in \text{Spec}_{\text{dis}}(\Delta):\lambda_i\leq\lambda\}| \) be the Weyl counting function. The author shows in several cases that \(N_d(\lambda)\) satisfies the Weyl estimate: \[ \lim_{\lambda\rightarrow\infty}\sup N_d(\lambda)\lambda^{-n/2} \leq(4\pi)^{-n/2}\text{vol}(M)(\Gamma(n/2+1)^{-1}\dim E_\sigma. \]
scientific article; zbMATH DE number 1465058