Szegö type limit theorems on the Heisenberg group

2019/03/04 by Mondal, Shyam Swarup, Swain, Jitendriya
#FOS: Mathematics #Functional Analysis (math.FA) #Heisenberg group #Hermite operator #Pseudo-differential operator #Tauberian theorem

paper · doi:10.48550/arxiv.1903.01163

Abstract

Let H=-Δ_ℍ+V be the Schrödinger operator on the Heisenberg group ℍⁿ, where Δ_ℍ is the full laplacian on ℍⁿ and V is a positive smooth potential, bounded below and grows like |g|^κ, κ>0 for large |g|. Let P_r be the orthogonal projection of L²(ℍⁿ) onto the space of eigenfunctions of H with eigenvalue ≤ r; Let A be a 0-th order self-adjoint pseudo-differential operator on L²(ℍⁿ) relative to the operator 1+|λ|H+V(g), g∈ ℍⁿ, λ∈ ℝ^* with symbol a(g, λ), where H is the Hermite operator on L²(ℝⁿ) then lim_r→∞ \fractr~f(P_rAP_r)tr~(P_r) amp;= lim_r→∞ \frac∫_G^rf(a_{g, λ}(ξ, x)) dξ dx dg dμ(λ) ∫_G^r dξ dx dg dμ(λ), (Assuming one limit exists) where G^r=\(g, λ, ξ, x)∈ ℍⁿ × ℝ^*× ℝⁿ× ℝⁿ : |λ|(1+|ξ| ²+|x|²)+V(g)≤ r \, a(g, λ)=Op^W(a_{g, λ}), and μ(λ) is the Plancherel measure on the Heisenberg group. Also we show that the above limit on the right hand side remains unaltered under a compact perturbation of the pseudo-differential operator A or a perturbation of the Schrödinger operator H by bounded self-adjoint operators on L²(ℍⁿ).

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