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 $\langle \phi_{el} \otimes \psi_{ph} | H_{\alpha, \Lambda} (\phi_{el} \otimes \psi_{ph}) \rangle$, where $H_{\alpha, \Lambda}$ is the corresponding Hamiltonian with fine structure constant $\alpha >0$ and ultraviolet cutoff $\Lambda < \infty$, and $\phi_{el}$ and $\psi_{ph}$ are normalized electron and photon wave functions, respectively. Lieb and Loss showed that $c \alpha^{1/2} \Lambda^{3/2} \leq E_{\alpha, \Lambda} \leq c^{-1} \alpha^{2/7} \Lambda^{12/7}$ 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_{\alpha, \Lambda}}{F[1] \, \alpha^{2/7} \, \Lambda^{12/7}} - 1 \bigg| \ \leq \ C \, \alpha^{4/105} \, \Lambda^{-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 $\Lambda \to \infty$.