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.
Related
- Computation of leading coefficients in asymptotics of relative quantum invariants
- Sharp universal death of entanglement threshold for Pauli Hamiltonians
- Sharp Minimax Gram-Determinant Asymptotics for Labelled Unit Decompositions of the Identity and Recent Developments on Makai's Conjecture
- Asymptotics of Laplace and Stokes solutions in the gap between a sharp tip and a smooth boundary
- Sharp asymptotics for a transport model with a nonlocal condition of the Fisher-KPP type at the boundary
BibTeX
@misc{futakuchi2026sharp,
title = {{The sharp leading asymptotics of the self-energy of the spinless Pauli-Fierz model}},
author = {Shinichiro Futakuchi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34122},
eprint = {2610.34122},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34122},
}
RIS
TY - UNPB
TI - The sharp leading asymptotics of the self-energy of the spinless Pauli-Fierz model
AU - Shinichiro Futakuchi
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34122
UR - https://arxiv.org/abs/2610.34122
AB - 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.
ER -