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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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