On the Ultraviolet Limit of the Pauli–Fierz Hamiltonian in the Lieb–Loss Model

Abstract

Two decades ago, Lieb and Loss (Self-energy of electrons in non-perturbative QED. Preprint arXiv:math-ph/9908020 and mp-arc #99–305, 1999) approximated the ground state energy of a free, nonrelativistic electron coupled to the quantized radiation field by the infimum E_{alpha , Lambda } of all expectation values langle phi _{el} otimes psi _{ph} | H_{alpha , Lambda } (phi _{el} otimes psi _{ph}) rangle , where H_{alpha , Lambda } is the corresponding Hamiltonian with fine structure constant alpha >0 and ultraviolet cutoff Lambda < infty , and phi _{el} and psi _{ph} are normalized electron and photon wave functions, respectively. Lieb and Loss showed that c alpha ^{1/2} Lambda ^{3/2} le E_{alpha , Lambda } le c^{-1} alpha ^{2/7} Lambda ^{12/7} for some constant c >0. In the present paper, we prove the existence of a constant C < infty , such that |Eα,ΛF1α2/7Λ12/7-1|≤Cα4/105Λ-4/105\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\bigg | \\frac{E_{\\alpha , \\Lambda }}{F_1 \\, \\alpha ^{2/7} \\, \\Lambda ^{12/7}} - 1 \\bigg | \\ \\le \\ C \\, \\alpha ^{4/105} \\, \\Lambda ^{-4/105} \\end{aligned}$$\\end{document}holds true, where F_1 >0 is an explicit universal number. This result shows that Lieb and Loss’ upper bound is actually sharp and gives the asymptotics of E_{alpha , Lambda } uniformly in the limit alpha rightarrow 0 and in the ultraviolet limit Lambda rightarrow infty .