Gaussian decay for a difference of traces of the Schrödinger semigroup associated to the isotropic harmonic oscillator.
Résumé
This paper deals with the derivation of a sharp estimate on the difference of traces of the one-parameter Schrödinger 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 Schrödinger 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 $\Lambda_{L}^{d}:= \{\bold{x} \in \mathbb{R}^{d}:-L/2 < x_{l} < L/2,\,l=1,\ldots,d\}$, 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 a Gaussian decay in $L$ for $L$ sufficiently large. The estimate we derive is sharp in the sense that its behavior when $\kappa \downarrow 0$ and $t \downarrow 0$ is similar to the one given by $\mathrm{Tr}_{L^{2}(\mathbb{R}^{d})}\mathrm{e}^{-t H_{\infty,\kappa}} = (2\sinh( \frac{\kappa}{2}t))^{-d}$. Further, we give a simple application within the framework of quantum statistical mechanics.
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