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

Beau, Mathieu and Savoie, Baptiste (2014) Gaussian decay for a difference of traces of the Schrödinger semigroup associated to the isotropic harmonic oscillator. (Preprint)

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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_∞,κ the self-adjoint realization in L 2 (R d ), d ∈ {1, 2, 3} of the Schrödinger operator −(1/2)∆ + (1/2)κ^2*|x|^2, κ > 0 and by H_L,κ, L > 0 the Dirichlet realization in L^2(Λ^d_L) where Λ^d_L := {x ∈ R^d : −L/2 < x_l < L/2, l = 1, . . . , d}, we prove that the difference of traces Tr_(L^2(R^d))*e^(−tH_∞,κ) − Tr_(L^2(Λ^d_L)*e^(−tH_L,κ), t > 0 has a Gaussian decay in L for L sufficiently large. L The estimate we derive is sharp in the sense that its behavior when κ ↓ 0 and t ↓ 0 is similar to the one given by Tr_(L^2(R^d))*e^(−tH_∞,κ) = (2sinh((κ/2)*t))^(−d). Further, we give a simple application within the framework of quantum statistical mechanics.

Item Type: Article
Uncontrolled Keywords: Quantum harmonic oscillator, Gibbs semigroups, Mehler’s formula, Duhamel-like formula, Geometric perturbation theory
Divisions: School of Theoretical Physics > Preprints
Date Deposited: 19 Jun 2018 13:48
Last Modified: 12 Jul 2018 11:51
URI: http://dair.dias.ie/id/eprint/533

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