Localization of the number of photons of ground states in nonrelativistic QED

One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L²(ℝ³) ⊗ F ≅ L² (ℝ³; F), where F is the Boson Fock space over L²(ℝ³ × {1, 2}). It is shown that the ground state, ψ_g, of H belongs to ∩_{k = 1}^∞D(1 ⊗ N^k), where N denotes the number operator of F. Moreover, it is shown that for almost every electron position variable x ∈ ℝ³ and for arbitrary k ≥ 0, ∥(1 ⊗ N^k/2ψ_g(x)∥_F ≤ D_ke^-δ|x|m+1 with some constants m ≥ 0, D_k > 0, and δ > 0 independent of k. In particular ψ_g ∈ ∩_{k = 1}^∞D(e^β|x|m+1 ⊗ N^k) for 0 < 0 < δ/2 is obtained.