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For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, weprove that if there exists a family of quasimodes (as is the case when theexterior of the obstacle has stable trapped rays), then there exist near-zeroeigenvalues of the standard variational formulation of the exterior Dirichletproblem (recall that this formulation involves truncating the exterior domainand applying the exterior Dirichlet-to-Neumann map on the truncation boundary).  Our motivation for proving this result is that a) the finite-element methodfor computing approximations to solutions of the Helmholtz equation is based onthe standard variational formulation, and b) the location of eigenvalues, andespecially near-zero ones, plays a key role in understanding how iterativesolvers such as the generalised minimum residual method (GMRES) behave whenused to solve linear systems, in particular those arising from thefinite-element method. The result proved in this paper is thus the first steptowards rigorously understanding how GMRES behaves when applied todiscretisations of high-frequency Helmholtz problems under strong trapping (thesubject of the companion paper [Marchand, Galkowski, Spence, Spence, 2021]).

Eigenvalues of the truncated Helmholtz solution operator under strong trapping