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

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 $E_{\alpha ,\mathrm{\Lambda }}$ 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 $\mathrm{\Lambda }<\mathrm{\infty }$, and ${\varphi }_{el}$ and ${\psi }_{ph}$ are normalized electron and photon wave functions, respectively. Lieb and Loss showed that $c \alpha^{1/2} \Lambda^{3/2} \leq E_{\alpha, \Lambda} \leq c^{-1} \alpha^{2/7} \Lambda^{12/7}$forsomeconstant$c >0$. In the present paper we prove the existence of a constant $C<\mathrm{\infty }$, 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 $E_{\alpha ,\mathrm{\Lambda }}$ uniformly in the limit $\alpha \to 0$ and in the ultraviolet limit $\mathrm{\Lambda }\to \mathrm{\infty }$.