Let Ω be a Euclidean domain with smooth boundary, and let λ be a positive real number.
The Dirichlet-to-Neumann operator at frequency λ, denoted R(λ), maps smooth functions on the boundary of Ω to smooth functions on the boundary of Ω. It is defined as follows: if f∈C∞(∂Ω), then we find the function u defined in the interior of Ω satisfying the Helmholtz equation (-Δ-λ2)u=0, and with boundary value f. This can be solved uniquely provided that λ2 is not a Dirichlet eigenvalue of -Δ. Then R(λ)f is defined to be the normal derivative of u at ∂Ω.
The operator R(λ) is intimately related to eigenvalue counting functions for the domain Ω. For example, the number of negative eigenvalues of R(λ) is equal to the difference between the Neumann and Dirichlet eigenvalue counting functions (at eigenvalue λ2).
In this talk, I will explain how to get a leading asymptotic for the number of eigenvalues of R(λ) in the interval [aλ,bλ) as λ→∞, given a condition on the periodic billiard trajectories in Ω.