Ground states and associated path measures in the renormalized Nelson model

Abstract

We prove the existence, uniqueness, and strict positivity of ground states of the possibly massless renormalized Nelson operator under an infrared regularity condition and for Kato decomposable electrostatic potentials fulfilling a binding condition. If the infrared regularity condition is violated, then we show non-existence of ground states of the massless renormalized Nelson operator with an arbitrary Kato decomposable potential. Furthermore, we prove the existence, uniqueness, and strict positivity of ground states of the massless renormalized Nelson operator in a non-Fock representation where the infrared condition is unnecessary. Exponential and superexponential estimates on the pointwise spatial decay and the decay with respect to the boson number for elements of spectral subspaces below localization thresholds are provided. Moreover, some continuity properties of ground state eigenvectors are discussed. Byproducts of our analysis are a hypercontractivity bound for the semigroup and a new remark on Nelson's operator theoretic renormalization procedure. Finally, we construct path measures associated with ground states of the renormalized Nelson operator. Their analysis entails improved boson number decay estimates for ground state eigenvectors, as well as upper and lower bounds on the Gaussian localization with respect to the field variables in the ground state. As our results on uniqueness, positivity, and path measures exploit the ergodicity of the semigroup, we restrict our attention to one matter particle. All results are non-perturbative.

72 pages. Second version as published in Reviews in Mathematical Physics; contains a few minor revisions to improve readability and updated discussions of the related literature

References in corpus (7)

• Asymptotic Completeness in Quantum Field Theory: Translation Invariant Nelson Type Models Restricted to the Vacuum and One-Particle Sectors
• On Nelson-type Hamiltonians and abstract boundary conditions
• On the domain of the Nelson Hamiltonian
• On the self-adjointness of H+A*+A
• Absence of Ground States in the Renormalized Massless Translation-Invariant Nelson Model
• The resolvent of the Nelson Hamiltonian improves positivity
• Feynman-Kac formulas for Dirichlet-Pauli-Fierz operators with singular coefficients

Cited by in corpus (7)

• Ground State Properties in the Quasi-Classical Regime
• Absence of Ground States in the Renormalized Massless Translation-Invariant Nelson Model
• Bogoliubov Dynamics and Higher-order Corrections for the Regularized Nelson Model
• Non-Fock Ground States in the Translation-Invariant Nelson Model Revisited Non-Perturbatively
• A Lower Bound on the Critical Momentum of an Impurity in a Bose-Einstein Condensate
• Feynman-Kac formula and asymptotic behavior of the minimal energy for the relativistic Nelson model in two spatial dimensions
• Feynman-Kac formulas for Dirichlet-Pauli-Fierz operators with singular coefficients