Two-sided bounds on the point-wise spatial decay of ground states in the renormalized Nelson model with confining potentials

TL;DR

This paper establishes two-sided bounds on how quickly ground states decay spatially in the renormalized Nelson model with confining potentials, linking decay rates to the Agmon distance, and matching previous upper bounds asymptotically.

Contribution

It provides the first rigorous two-sided bounds on the spatial decay of ground states in the Nelson model with confining potentials, connecting decay rates to the Agmon distance.

Findings

Lower bounds on decay rates derived using Feynman-Kac representations.

Decay bounds match asymptotically with previous upper bounds.

Decay characterized by the Agmon distance from Schrödinger operator analysis.

Abstract

We study the renormalized Nelson model for a scalar matter particle in a continuous confining potential interacting with a possibly massless quantized radiation field. When the radiation field is massless we impose a mild infrared regularization ensuring that the Nelson Hamiltonian has a non-degenerate ground state in all considered cases. Employing Feynman-Kac representations, we derive lower bounds on the point-wise spatial decay of the partial Fock space norms of ground state eigenvectors. Here the exponential rate function governing the decay is given by the Agmon distance familiar from the analysis of Schr\"{o}dinger operators. For a large class of confining potentials, our lower bounds on the decay of ground state eigenvectors match asymptotically with the upper bounds implied by previous work of the present authors.