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

Abstract

AbstractTwo decades ago, Lieb and Loss (Self-energy of electrons in non-perturbative QED. PreprintarXiv:math-ph/9908020and mp-arc #99–305, 1999) approximated the ground state energy of a free, nonrelativistic electron coupled to the quantized radiation field by the infimum$$E_{\alpha , \Lambda }$$Eα,Λof all expectation values$$\langle \phi _{el} \otimes \psi _{ph} | H_{\alpha , \Lambda } (\phi _{el} \otimes \psi _{ph}) \rangle $$⟨ϕel⊗ψph|Hα,Λ(ϕel⊗ψph)⟩, where$$H_{\alpha , \Lambda }$$Hα,Λis the corresponding Hamiltonian with fine structure constant$$\alpha >0$$α>0and ultraviolet cutoff$$\Lambda < \infty $$Λ<∞, and$$\phi _{el}$$ϕeland$$\psi _{ph}$$ψphare normalized electron and photon wave functions, respectively. Lieb and Loss showed that$$c \alpha ^{1/2} \Lambda ^{3/2} \le E_{\alpha , \Lambda } \le c^{-1} \alpha ^{2/7} \Lambda ^{12/7}$$cα1/2Λ3/2≤Eα,Λ≤c-1α2/7Λ12/7for some constant$$c >0$$c>0. In the present paper, we prove the existence of a constant$$C < \infty $$C<∞, such that$$\begin{aligned} \bigg | \frac{E_{\alpha , \Lambda }}{F_1 \, \alpha ^{2/7} \, \Lambda ^{12/7}} - 1 \bigg | \ \le \ C \, \alpha ^{4/105} \, \Lambda ^{-4/105} \end{aligned}$$|Eα,ΛF1α2/7Λ12/7-1|≤Cα4/105Λ-4/105holds true, where$$F_1 >0$$F1>0is an explicit universal number. This result shows that Lieb and Loss’ upper bound is actually sharp and gives the asymptotics of$$E_{\alpha , \Lambda }$$Eα,Λuniformly in the limit$$\alpha \rightarrow 0$$α→0and in the ultraviolet limit$$\Lambda \rightarrow \infty $$Λ→∞.