The first order expansion of a ground state energy of the $\phi^4$ model with cutoffs

TL;DR

This paper derives the first order expansion of the ground state energy for the $^4$ quantum field model with spatial and momentum cutoffs, establishing a rigorous mathematical framework for its spectral analysis.

Contribution

It provides the first order expansion of the ground state energy of the $^4$ model with cutoffs, under regularity conditions, using a self-adjoint Hamiltonian framework.

Findings

First order energy expansion derived for the $^4$ model with cutoffs

Hamiltonian shown to be self-adjoint on boson Fock space

Results depend on regularity conditions of the momentum cutoff

Abstract

In this paper, we investigate the $\phi^4$ model with cutoffs. By introducing a spatial cutoff and a momentum cutoff, the total Hamiltonian is a self-adjoint operator on a boson Fock space. Under regularity conditions of the momentum cutoff, we obtain the first order expansion of a non-degenerate ground state energy of the total Hamiltonian.