The sharp leading asymptotics of the self-energy of the spinless Pauli-Fierz model

Abstract

We consider the ground state energy E_0(alpha, Lambda) of a single spinless non-relativistic electron coupled to the quantized radiation field with a sharp ultraviolet cutoff Lambda (the Pauli-Fierz model), with fine structure constant alpha = e^2/(4 pi). Lieb and Loss proved that C_1 alpha^(1/2) Lambda^(3/2) <= E_0 <= C_2 alpha^(2/7) Lambda^(12/7) for large Lambda and suggested that the upper bound, which comes from product states, gives the correct order. We prove that, for every fixed alpha > 0, E_0(alpha, Lambda) = (1+o(1)) c_alpha alpha^(2/7) Lambda^(12/7) as Lambda -> infinity, with an explicit constant c_alpha = 0.99338...; more generally, this holds uniformly in the joint limit Lambda/alpha -> infinity, alpha^(5/2) Lambda -> infinity, provided log(2+alpha) = o((Lambda/alpha)^(1/21)). Since our upper bound holds already for the infimum over product states, it follows that the ratio of E_0 to this infimum tends to one: electron-field correlations do not contribute to the leading term. The condition alpha^(5/2) Lambda -> infinity cannot be dropped, since for alpha^(5/2) Lambda -> 0 the vacuum trial state gives a smaller energy. The upper bound follows from a coherent state bound for the field energy in a fixed electron density. The lower bound combines a periodization of the field, a Born-Oppenheimer bound in which phase multiplication and a Hardy inequality in field space are applied block by block, and a lower bound on the magnetic Neumann Laplacian on a cube for band-limited fields; the key point of the last is that a band-limited field obeying a weighted gradient condition cannot concentrate its vorticity.