Some upper bounds on the number of resonances for manifolds with infinite cylindrical ends (Q1854634)

Let \(M\) be a manifold with infinite cylindrical ends. The Laplace-Beltrami operator \(\Delta_M\) has for spectral parameter space a Riemann surface \(\widehat Z_M\), an infinite cover of \(\mathbb C\) containing \(\mathbb C\setminus\mathbb R\) as physical sheet: the resolvent distributional kernel \((\Delta_M-z)^{-1}(m,m')\), \(z\in\mathbb C\setminus\mathbb R^+\), \(m,m'\in M\), admits a meromorphic extension to \(\widehat Z_M\). Beyond \(L^2\)-eigenvalues, resolvent poles define the resonance set. The main topic of this paper it to prove various upper bounds on the number of resonances located in domains \((\Omega_m)\) of \(\widehat Z_M\). Techniques used by the author are adaptation of those developped by \textit{R. Melrose} [Journ. ``Équations Dériv. Partielles'', St. Jean-De-Monts 1984, Conf. No. 3, 8 p. (1984; Zbl 0621.35073)], \textit{M. Zworski} [J. Funct. Anal. 82, 370-403 (1989; Zbl 0681.47002)] and \textit{G. Vodev} [Commun. Math. Phys. 146, 205-216 (1992; Zbl 0766.35032)]. With a degeneracy hypothesis on the spectrum \(\Sigma_M\), the branching points of \(\widehat Z_M\) are separated sufficiently, so that the author can define large domains \(\Omega_m\) as union of disjoints disks, where the Jensen theorem is uniformly used. The number of resonances in \(\Omega_m\) is \({\mathcal O}(m^{k})\) when \(m\to\infty\), where \(k=\dim M- 1\) or \(\dim M- 2\), depending on the choice of \(\Omega_m\). The paper contains also precise analysis how scattering poles coincide with resolvent poles (\(L^2\)-eigenvalue and resonances), multiplicities included. Waveguides in mesoscopy physics enter in the framework studied by the author and are very popular today [see e.g. \textit{J. Edward}, J. Math. Anal. Appl. 272, 89-116 (2002; Zbl 1119.78317)].