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Locating resonances on hyperbolic cones. (English) Zbl 1426.35091

Let \(\mathcal{Y}:=(Y,h)\) be a closed Riemannian manifold of dimension \(n\). Let \(\mathcal{X}:=(\mathbb{R}^+\times Y, g:=dr^2+\sinh^2(r)\cdot h)\) be the associated hyperbolic cone; \(X\) has an isolated conic singularity at \(r=0\) except in the special case that \(\mathcal{Y}\) is hyperbolic space. Let \(\Delta_g\) be the scalar Laplacian. The resolvant \[ R(\lambda):=\left(-\Delta_g-\lambda^2-\frac{n^2}4\right)^{-1} \] is a bounded operator on \(L^2(x,g)\) for \(\operatorname{Im}(\lambda)>0\) with the possible exception of finitely many poles; \(R(\lambda)\) admits a meromorphic extension to \(\mathbb{C}\). The poles of this extension are the resonances. The authors explicitly compute these resonances in terms of the eigenvalues of \(\mathcal{Y}\) using a separation of variables and Kummer’s connection formula for hypergeometric functions. The authors also derive a Weyl’s law for \(\mathcal{X}\) assuming that \(\mathcal{Y}\) is generic. The authors use the warped product structure to give a representation of the resolvant.

Section 1 presents an introduction to the matter. An explicit expression for the resolvent is given in Section 2. In Section 3, the resonances are located using the structure of the hypergeometric and Gamma functions which appear in the resolvent. The final section presents examples. Hyperbolic space is examined and resonances in the presence of a conic singularity are given.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
58J05 Elliptic equations on manifolds, general theory
47A10 Spectrum, resolvent
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