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

Abstract

Two decades ago, Lieb and Loss proposed to approximate 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 $⟨{\varphi }_{el}\otimes {\psi }_{ph}|H_{\alpha ,\Lambda }({\varphi }_{el}\otimes {\psi }_{ph})⟩$, where $H_{\alpha ,\Lambda }$ is the corresponding Hamiltonian with fine structure constant $\alpha >0$ and ultraviolet cutoff $\Lambda <\infty$, and ${\varphi }_{el}$ 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 \begin{align*} \bigg| \frac{E_{α, Λ}}{F[1] \, α^{2/7} \, Λ^{12/7}} - 1 \bigg| \ \leq \ C \, α^{4/105} \, Λ^{-4/105} \end{align*} 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 $\alpha \to 0$ and in the ultraviolet limit $\Lambda \to \infty$.