Gaussian decay for a difference of traces of the Schr\"odinger semigroup associated to the isotropic harmonic oscillator

This paper deals with the derivation of a sharp estimate on the difference of traces of the one-parameter Schr\"odinger semigroup associated to the quantum isotropic harmonic oscillator. Denoting by $H_{\infty,\kappa}$ the self-adjoint realization in $L^{2}(\mathbb{R}^{d})$, $d \in \{1,2,3\}$ of the Schr\"odinger operator $-\frac{1}{2} \Delta + \frac{1}{2} \kappa^{2}\vert \bold{x}\vert^{2}$, $\kappa>0$ and by $H_{L,\kappa}$, $L>0$ the Dirichlet realization in $L^{2}(\Lambda_{L}^{d})$ where $\Lambda_{L}^{d}:=\{\bold{x} \in \mathbb{R}^{d}:- \frac{L}{2} < x_{l} < \frac{L}{2},\,l=1,\ldots,d\}$, we prove that the difference of traces $\mathrm{Tr}_{L^{2}(\mathbb{R}^{d})} \mathrm{e}^{-t H_{\infty,\kappa}} - \mathrm{Tr}_{L^{2}(\Lambda_{L}^{d})}\mathrm{e}^{-t H_{L,\kappa}}$, $t>0$ 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 $t \downarrow 0$ is similar to the one given by $\mathrm{Tr}_{L^{2}(\mathbb{R}^{d})}\ mathrm{e}^{-t H_{\infty,\kappa}} = (2\sinh( \frac{\kappa}{2}t))^{-d}$ and the exponential decay in $t$ arising from $\mathrm{Tr}_{L^{2}(\mathbb{R}^{d})}\mathrm{e}^{-t H_{\infty,\kappa}}$ when $t\uparrow \infty$ is preserved. For illustrative purposes, we give a simple application within the framework of quantum statistical mechanics.