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Spectral asymptotics of the Laplace operator on manifolds with cylindrical ends. (English) Zbl 0842.58074

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.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
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