Parnovski, L. B. Spectral asymptotics of the Laplace operator on manifolds with cylindrical ends. (English) Zbl 0842.58074 Int. J. Math. 6, No. 6, 911-920 (1995). The author deals with the spectral asymptotics of a manifold \(M\) with cylindrical ends, i.e., with the asymptotics of the sum \(N_d(\lambda) + N_c(\lambda)\), where \(N_d(\lambda)\) and \(N_c(\lambda)\) are correspondingly the counting functions of the discrete and continuous spectra of the positive Laplace operator (with Dirichlet or Neumann boundary conditions) on \(M\). The counting function of the continuous spectrum is defined in this paper as \[ N_c(\lambda) := {i \over 2\pi} \int^\lambda_{\mu_1} d \log \text{det } T(\nu), \] \(T(\nu)\) being the scattering matrix and \(\mu_1\) is the first eigenvalue of the cylinder’s section. The main result of this paper is a proof of the following asymptotic formula: \[ N_d(\lambda) + N_c(\lambda) = C_{n + 1} |M_0|\lambda^{(n + 1)/2} + O(\lambda^{n/2}), \] where \(|M_0|\) is the regularized volume of \(|M|\), and \(C_n = (2\pi)^{-n} \omega_n\) is the Weyl constant, \(\omega_n\) being the volume of a unit ball in \(\mathbb{R}^n\). This formula is proved by using the modification of the Colin de Verdière cut-off Laplacian. It is also valid for manifolds with cusps of rank one. Reviewer: S.Nikčević (Beograd) Cited in 14 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 35P20 Asymptotic distributions of eigenvalues in context of PDEs 35J10 Schrödinger operator, Schrödinger equation Keywords:manifolds with cylindrical ends; spectral asymptotics; Laplace operator PDFBibTeX XMLCite \textit{L. B. Parnovski}, Int. J. Math. 6, No. 6, 911--920 (1995; Zbl 0842.58074) Full Text: DOI