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Spectral asymptotics of Laplace operators on surfaces with cusps. (English) Zbl 0849.35093
Let \(M\) be a compact perturbation of a surface with constant negative curvature, finite volume and \(k\) cusps, \(\varphi\) the determinant of the scattering matrix of the positive Laplace operator on \(M\), \(\{\lambda_j: j\in \mathbb{N}\}\) the sequence of eigenvalues, \[ N_d(T)= \sum_{\lambda_j< T^2} 1,\;N_c(T)\equiv - {1\over 4\pi} \int^T_{- T} {\varphi'\over \varphi}\Biggl({1\over 2}+ ir\Biggr) dr,\;N_p(T)= \sum_{\rho\in R, |\rho- {1\over 2}|< T} m(\rho), \] where \(R\) is the set of poles of \(\varphi\) and \(m(\rho)\) is the order of the pole \(\rho\).
Then \(N_d(T)+ N_c(T)= {|M|\over 4\pi} T^2- {k\over \pi} T\ln T+ O(T)\), \(N_c(T)- {1\over 2} N_p(T)= o(T^{{3\over 2}+ \varepsilon})\) for all \(\varepsilon> 0\) and \(N_c(T)- {1\over 2} N_p(T)= o(T^{1+ \varepsilon})\) for all \(\varepsilon> 0\) if \(|\text{Re } \rho|< C\) for all \(\rho\in \mathbb{R}\). If the Liouville measure of the periodic trajectories of the geodesic flow on \(M\) equals zero, then \(N_d(T)+ N_c(T)= {|M|\over 4\pi} T^2- {k\over \pi} T\ln T+ {k\over \pi} (1- \ln 2) T+ o(T)\).
Reviewer: G.Bottaro (Genova)

MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
47F05 General theory of partial differential operators
58J05 Elliptic equations on manifolds, general theory
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