A lower bound on the renormalized Nelson model

Gonzalo A. Bley

doi:10.1063/1.5008831

A lower bound on the renormalized Nelson model

Gonzalo A. Bley^*

^*Corresponding author af dette arbejde

Institut for Matematik

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

Abstract

We provide explicit lower bounds for the ground-state energy of the renormalized Nelson model in terms of the coupling constant α and the number of particles N, uniform in the meson mass and valid even in the massless case. In particular, for any number of particles N and large enough α, we provide a bound of the form -Cα²N³ log²(αN), where C is an explicit positive numerical constant; and if α is sufficiently small, we give one of the form -Cα²N³ log² N for N ≥ 2 and -Cα² for N = 1. Whereas it is known that the renormalized Hamiltonian of the Nelson model is bounded below (as realized by Nelson) and implicit lower bounds have been given elsewhere (as in a recent work by Gubinelli, Hiroshima, and Lörinczi), ours seem to be the first fully explicit lower bounds with a reasonable dependence on α and N. We emphasize that the logarithmic term in the bounds above is probably an artifact in our calculations since one would expect that the ground-state energy should behave as -Cα²N³ for large N or α, as in the polaron model of Fröhlich.

Originalsprog	Engelsk
Artikelnummer	061901
Tidsskrift	Journal of Mathematical Physics
Vol/bind	59
Nummer	6
ISSN	0022-2488
DOI	https://doi.org/10.1063/1.5008831
Status	Udgivet - 1 jun. 2018

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10.1063/1.5008831

http://arxiv.org/pdf/1609.08590

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http://www.scopus.com/inward/record.url?scp=85048757877&partnerID=8YFLogxK

Semiclassical Quantum Mechanics
Fournais, S. (PI), Madsen, P. (Deltager), Mikkelsen, S. (Deltager), Miqueu, J.-P. C. (Deltager) & Bley, G. (Deltager)
01/07/2015 → 31/12/2020
Projekter: Projekt › Forskning

Citationsformater

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AB - We provide explicit lower bounds for the ground-state energy of the renormalized Nelson model in terms of the coupling constant α and the number of particles N, uniform in the meson mass and valid even in the massless case. In particular, for any number of particles N and large enough α, we provide a bound of the form -Cα2N3 log2(αN), where C is an explicit positive numerical constant; and if α is sufficiently small, we give one of the form -Cα2N3 log2 N for N ≥ 2 and -Cα2 for N = 1. Whereas it is known that the renormalized Hamiltonian of the Nelson model is bounded below (as realized by Nelson) and implicit lower bounds have been given elsewhere (as in a recent work by Gubinelli, Hiroshima, and Lörinczi), ours seem to be the first fully explicit lower bounds with a reasonable dependence on α and N. We emphasize that the logarithmic term in the bounds above is probably an artifact in our calculations since one would expect that the ground-state energy should behave as -Cα2N3 for large N or α, as in the polaron model of Fröhlich.

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A lower bound on the renormalized Nelson model

Abstract

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Fingeraftryk

Projekter

Semiclassical Quantum Mechanics

Citationsformater