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A lower bound on the renormalized Nelson model

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  • Gonzalo A. Bley

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.

TidsskriftJournal of Mathematical Physics
StatusUdgivet - 1 jun. 2018

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