Gaussian decay for a difference of traces of the Schrödinger semigroup associated to the isotropic harmonic oscillator

Abstract

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 $-\frac{1}{2}\Delta +\frac{1}{2}{\kappa }^2|\mathbf{x}{|}^2$, $\kappa >0$ and by $H_{L,\kappa }$, $L>0$ the Dirichlet realization in $L^2({\Lambda }_L^d)$ where ${\Lambda }_L^d:=\{\mathbf{x}\in {\mathbb{R}}^d:-\frac{L}{2}<x_l<\frac{L}{2},l=1,\dots ,d\}$, 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.

arXiv commentThis paper somehow corresponds to a part of the former version of the article 'The ideal Bose gas in open harmonic trap systems and its condensation revisited', arXiv:1306.3639v1. Accepted for publication in JMAA