Gaussian decay for a difference of traces of the Schrödinger 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 L2(Rd), d∈{1,2,3} of the Schrödinger operator −12Δ+12κ2|x|2, κ>0 and by HL,κ, L>0 the Dirichlet realization in L2(ΛLd) where ΛLd:={x∈Rd:−L20 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 TrL2(Rd)e−tH∞,κ=(2sinh(κ2t))−d and the exponential decay in t arising from TrL2(Rd)e−tH∞,κ when t↑∞ is preserved. For illustrative purposes, we give a simple application within the framework of quantum statistical mechanics.