Spectral asymptotics with a remainder estimate of the Neumann Laplacian on horns: the case of the rapidly growing counting function

Abstract

We study the Neumann Laplacian in unbounded regions of the form Ω = {(t, x) | t >O,f(t)−1x ∊ Ω′}, where Ω′ ⊂ ℝn−1 is a bounded open set with the Lipschitz boundary and f decays in such a way that the spectrum of is discrete but the counting function N(λ, ) of the spectrum grows faster than a power of λ, a typical example being f(t) = exp (– t In … In t), for t ≧ t0. We compute the principal term of the asymptotics of N(λ, ), with a remainder estimate.