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Spectral estimates for towers of noncompact quotients. (English) Zbl 0972.11045

The Weyl law gives the asymptotic \(\#\{\lambda_i\leq\lambda\}\sim c_d\text{ vol}(M)\lambda^{d/2}\) for the eigenvalue spectrum \(\lambda_1\leq \lambda_2\leq\ldots\) of the Laplace-Beltrami operator on a compact Riemannian manifold \(M\) of dimension \(d\), the constant \(c_d\) depending only on the dimension \(d\) of \(M\). The authors establish upper asymptotic estimates in case of a non-compact (but of finite volume) locally symmetric space \(M=X/\Gamma\) (\(X\) is a symmetric space of non compact type, \(\Gamma\) a discrete subgroup of the isometry group \(G\) of \(X\)). In this case, the Laplace-Beltrami has also continuous spectrum, but, if restricted to cuspidal functions, it has purely discrete spectrum \(\sigma_{\text{cusp}}\) for which the authors prove \[ \#\{\lambda_i\leq\lambda,\lambda_i\in\sigma_{\text{cusp}}\}\leq C[\Gamma,\Gamma_0](1+ \lambda)^{d/2}. \] This estimate is uniform with respect to \(\Gamma\) provided a condition of bounded depth for the groups \(\Gamma\), which expresses that it is uniformly commensurable with a principal congruence arithmetic group. In the case of one single \(\Gamma\), H. Donnelly [J. Differ. Geom. 17, 239-253 (1982; Zbl 0494.58029)] has established this upper bound (which reduces to the classical Weyl law in the case of the modular space \(\text{PSL}(2,\mathbb Z)\setminus\mathbb H^2\)). The methods involve Neumann bracketing and heat kernel estimates, as used by Donnelly (ibid.) and W. Müller [Ann. Math. (2) 130, 473-529 (1989; Zbl 0701.11019); Geom. Funct. Anal. 8, 315-355 (1998; Zbl 1073.11514)], with an original construction of heat parametrix for domains with boundary conditions. A last section settles the main results in the adelic setting (which is important for the number theory point of view).
The uniform estimates will be used in a subsequent paper to study spectral results for towers (i.e., nested sequences \((\Gamma_j)\) of subgroups, with \(\Gamma_{j+1}\) of finite index in \(\Gamma_j\) and \(\bigcap_j\Gamma_j=\{1\}\)).

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
22E40 Discrete subgroups of Lie groups
53C35 Differential geometry of symmetric spaces
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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