A Semi-classical Trace Formula for Schrِdinger Operators in the Case of a Critical Energy Level

LetĤ=−(ℏ2/2) Δ+V(x) be a Schrödinger operator on Rn, with smooth potentialV(x)→+∞ as |x|→+∞. The spectrum ofĤis discrete, and one can study the asymptotic of the smoothed spectral densityϒ(E, ℏ)=∑k ϕ Ek(ℏ)−Eℏ,as ℏ→0. Here, {Ek(ℏ)}k∈Nis the spectrum ofĤand ϕ∈C∞0(R). We shall investigate the case whereEis a critical value of the symbolHofĤand, extending the work of Brummelhuis, Paul and Uribe in [3], we will prove the existence of a full asymptotic expansion forϒin terms ofℏand ln ℏ and compute the leading coefficient. We will consider new Weyl-type estimates for the counting function:NEc, ρ(ℏ)=#{k∈N/|Ek(ℏ)−E|⩽ρℏ}.