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 $\Lambda \to \infty$.
Categories: math-ph math.MP