Unstable States in a Model of Nonrelativistic Quantum Electrodynamics: Corrections to the Lorentzian Distribution

Abstract

We revisit the Lee-Friedrichs model as a model of atomic resonances in the hydrogen atom, using the dipole-moment matrix-element functions which have been exactly computed by Nussenzveig. The Hamiltonian H of the model is positive and has absolutely continuous spectrum. Although the return probability amplitude \(R_{\Psi }(t) = (\Psi , \exp (-iHt) \Psi )\) of the initial state \(\Psi \), taken as the so-called Weisskopf–Wigner (W.W.) state, cannot be computed exactly, we show that it equals the sum of an exponentially decaying term and a universal correction \(O(\beta ^{2}\frac{1}{t})\), for large positive times t and small coupling constants \(\beta \), improving on some results of King (Lett Math Phys 23:215–222, 1991). The remaining, non-universal, part of the correction is also shown to be of the same qualitative type. The method consists in approximating the matrix element of the resolvent operator operator in the W.W. state by a Lorentzian distribution. No use is made of complex energies associated to analytic continuations of the resolvent operator to ”physical” Riemann sheets. Other new results are presented, in particular a physical interpretation of the corrections, and the characterization of the so-called sojourn time \(\tau _{H}(\Psi )= \int _{0}^{\infty } |R_{\Psi }(t)|^{2} dt\) as the average lifetime of the decaying state, a standard quantity in (quantum) probability.