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 mp-arc #99–305, 1999) approximated the ground state energy a free, nonrelativistic electron coupled to quantized radiation field by infimum $$E_{\alpha , \Lambda }$$ Eα,Λ all expectation values $$\langle \phi _{el} \otimes \psi _{ph} | H_{\alpha } (\phi _{ph}) \rangle $$ xmlns:mml="http://www.w3.org/1998/Math/MathML">⟨ϕel⊗ψph|Hα,Λ(ϕel⊗ψph)⟩ where $$H_{\alpha xmlns:mml="http://www.w3.org/1998/Math/MathML">Hα,Λ is corresponding Hamiltonian with fine structure constant $$\alpha >0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">α>0 ultraviolet cutoff $$\Lambda < \infty xmlns:mml="http://www.w3.org/1998/Math/MathML">Λ<∞ $$\phi _{el}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">ϕel $$\psi _{ph}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">ψph are normalized photon wave functions, respectively. showed that $$c \alpha ^{1/2} ^{3/2} \le E_{\alpha c^{-1} ^{2/7} ^{12/7}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">cα1/2Λ3/2≤Eα,Λ≤c-1α2/7Λ12/7 for some xmlns:mml="http://www.w3.org/1998/Math/MathML">c>0 . In present paper, we prove existence $$C xmlns:mml="http://www.w3.org/1998/Math/MathML">C<∞ such $$\begin{aligned} \bigg \frac{E_{\alpha }}{F_1 \, ^{12/7}} - 1 \ C ^{4/105} ^{-4/105} \end{aligned}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">|Eα,ΛF1α2/7Λ12/7-1|≤Cα4/105Λ-4/105 holds true, $$F_1 xmlns:mml="http://www.w3.org/1998/Math/MathML">F1>0 an explicit universal number. This result shows Loss’ upper bound actually sharp gives asymptotics uniformly limit \rightarrow 0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">α→0 xmlns:mml="http://www.w3.org/1998/Math/MathML">Λ→∞