Construction of the ground state in nonrelativistic QED by continuous flows

Abstract

For a nonrelativistic hydrogen atom minimally coupled to the quantized radiation field we construct the ground state projection P gs by a continuous approximation scheme as an alternative to the iteration scheme recently used by Fröhlich, Pizzo, and the first author [V. Bach, J. Fröhlich, A. Pizzo, Infrared-finite algorithms in QED: The groundstate of an atom interacting with the quantized radiation field, Comm. Math. Phys. (2006), doi: 10.1007/s00220-005-1478-3]. That is, we construct P gs = lim t → ∞ P t as the limit of a continuously differentiable family ( P t ) t ⩾ 0 of ground state projections of infrared regularized Hamiltonians H t . Using the ODE solved by this family of projections, we show that the norm ‖ P ˙ t ‖ of their derivative is integrable in t which in turn yields the convergence of P t by the fundamental theorem of calculus.