Gaussian decay for a difference of traces of the Schrِdinger semigroup associated with the isotropic harmonic oscillator

This paper deals with the derivation of a sharp estimate on the difference of traces of the one-parameter Schrödinger semigroup associated with the quantum isotropic harmonic oscillator. Denoting by H ∞ , κ the self-adjoint realization in L 2 ( R d ) , d ∈ { 1 , 2 , 3 } of the Schrödinger operator − 1 2 Δ + 1 2 κ 2 | x | 2 , κ > 0 and by H L , κ , L > 0 the Dirichlet realization in L 2 ( Λ L d ) where Λ L d : = { x ∈ R d : − L 2 < x l < L 2 , l = 1 , … , d } , we prove that the difference of traces Tr L 2 ( R d ) e − t H ∞ , κ − Tr L 2 ( Λ L d ) e − t H L , κ , t > 0 has for L sufficiently large a Gaussian decay in L. Furthermore, the estimate that we derive is sharp in the two following senses: its behavior when t ↓ 0 is similar to the one given by Tr L 2 ( R d ) e − t H ∞ , κ = ( 2 sinh ( κ 2 t ) ) − d and the exponential decay in t arising from Tr L 2 ( R d ) e − t H ∞ , κ when t ↑ ∞ is preserved. For illustrative purposes, we give a simple application within the framework of quantum statistical mechanics.