On Nelson-type Hamiltonians and abstract boundary conditions

TL;DR

This paper develops a framework for defining and analyzing Nelson-type Hamiltonians with abstract boundary conditions, providing explicit domain characterizations for models involving nonrelativistic particles coupled to scalar bosons.

Contribution

It introduces a novel method to construct and characterize Hamiltonians using boundary conditions, applicable to both form-perturbed and renormalizable models.

Findings

Explicit domain characterization for Nelson-type Hamiltonians.

Unified treatment of form-perturbed and renormalizable models.

Application to models like Fröhlich's polaron and the massive Nelson model.

Abstract

We construct Hamiltonians for systems of nonrelativistic particles linearly coupled to massive scalar bosons using abstract boundary conditions. The construction yields an explicit characterisation of the domain of self-adjointness in terms of boundary conditions that relate sectors with different numbers of bosons. We treat both models in which the Hamiltonian may be defined as a form perturbation of the free operator, such as Fr\"ohlich's polaron, and renormalisable models, such as the massive Nelson model.