A Lower Bound on the Renormalized Nelson Model

2016/09/27 by Gonzalo A. Bley, Bley, Gonzalo A.
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1609.08590

arxiv created 2017/11/03 · arxiv updated 2017/11/06

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 -Cα² N³log²(αN), where C is an explicit positive numerical constant; and if α is sufficiently small, we give one of the form -Cα² N³log² 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 E. 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 -Cα² N³ for large N or α, as in the polaron model of H. Fröhlich.

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