Strong resolvent convergence of the scaled Pauli–Fierz Hamiltonians

Let $T_{1,\alpha}$ be the one-electron Hamiltonian

$$T_{1,\alpha}=\tfrac{1}{2}:\!\big(p-\sqrt{4\pi}\alpha^{3/2}A_{\alpha}(x)\big)^2\!:+H_{\mathrm{f}},$$

and let $T_{2,\alpha}$ be the corresponding Hamiltonian for two free electrons. Here $p$ (respectively, $p_1,p_2$) denotes the electron momentum, $A_{\alpha}$ is the ultraviolet-scaled quantized vector potential, $H_{\mathrm{f}}$ is the field Hamiltonian, and $a_0$ is the constant defined by

$$a_0=(2\pi)^2\left\langle A^+_1(0)\cdot A^+_1(0)\Omega,\left(\tfrac{1}{2}P_{\mathrm{f}}^2+H_{\mathrm{f}}\right)^{-1}A^+_1(0)\cdot A^+_1(0)\Omega\right\rangle.$$

Strong resolvent convergence conjecture. In the sense of strong convergence of resolvents,

$$\lim_{\alpha\to0}T_{1,\alpha}=\tfrac{1}{2}p^2+H_{\mathrm{f}}-a_0,$$

and

$$\lim_{\alpha\to0}T_{2,\alpha}=\tfrac{1}{2}p_1^2+\tfrac{1}{2}p_2^2+H_{\mathrm{f}}-2a_0.$$

The claim describes the singular weak-coupling limit in which the diverging norm of the coupling function produces finite self-energy shifts for one and two free electrons. The supplied text does not state whether this convergence has been proved or remains open.