For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary).
Analysis of PDEs Numerical Analysis Numerical Analysis 35J05, 35P15, 35B34, 35P25
We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Pad\'e approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition).
Numerical Analysis Numerical Analysis Analysis of PDEs 35J05, 65N99
Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $\Pi_\lambda(x, y)$ under the assumption that the set of geodesics that pass near both $x$ and $y$ has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions.
Analysis of PDEs Spectral Theory