Two-sided Bounds on the Point-Wise Spatial Decay of Ground States in the Renormalized Nelson Model with Confining Potentials

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 Lithner–Agmon distance familiar from the analysis of Schrödinger 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.