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 $H_{\mathrm{\infty },\kappa }$ the self-adjoint realization in $L^2({\mathbb{R}}^d)$, $d\in \{1,2,3\}$ of the Schr"odinger operator $-\frac{1}{2}\mathrm{\Delta }+\frac{1}{2}{\kappa }^2\mathrm{\mid }\mathbf{x}{\mathrm{\mid }}^2$, $\kappa >0$ and by $H_{L,\kappa }$, $L>0$ the Dirichlet realization in $L^2({\mathrm{\Lambda }}_L^d)$ 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 $$\mathrm{T}\mathrm{r}${L^{2}($\mathbb{R}$^{d})} $\mathrm{e}$^{-t H{\infty,\kappa}} - $\mathrm{T}\mathrm{r}${L^{2}(\Lambda{L}^{d})}$\mathrm{e}$^{-t H_{L,\kappa}}$,$t>0$ has for $L$ sufficiently large a Gaussian decay in $L$. Furthermore, the estimate that we derive is sharp in the two following senses: its behavior when $t \downarrow 0$issimilartotheonegivenby$$\mathrm{T}\mathrm{r}${L^{2}($\mathbb{R}$^{d})}\ mathrm{e}^{-t H{\infty,\kappa}} = (2\sinh( \frac{\kappa}{2}t))^{-d}$andtheexponentialdecayin$t$ arising from ${\mathrm{T}\mathrm{r}}_{L^2({\mathbb{R}}^d)}{\mathrm{e}}^{-t{H}_{\mathrm{\infty },\kappa }}$ when $t\uparrow \mathrm{\infty }$ is preserved. For illustrative purposes, we give a simple application within the framework of quantum statistical mechanics.