Spectral asymptotics for the semiclassical Dirichlet to Neumann operator

Let M be a compact Riemannian manifold with smooth boundary, and let R(λ) be the Dirichlet-to-Neumann operator at frequency X. The semiclassical Dirichlet-to-Neumann operator R_scl(λ) is defined to be λ^-1R(λ). We obtain a leading asymptotic for the spectral counting function for R_scl(λ) in an interval [a₁,a₂) as X -∗ oo, under the assumption that the measure of periodic billiards on T∗ M is zero. The asymptotic takes the form (Euqation presented) where κ(a) is given explicitly by (Euqation presented).