Analytic evaluation of exact transition matrix elements in nonrelativistic hydrogenic atoms

Long Time Behaviour of Unstable Multilevel Systems in Presence of Zero Energy Resonance

“…We also choose a special kind of the form factor that vanishes at zero energy with an integer power [15,16,21]. The result in [10] is only concerned with the short-range potential systems; however the method therein is developed in terms of the operators and thus it is applicable to our model by appropriate modifications in principle.…”

Zero energy resonance and the logarithmically slow decay of unstable multilevel systems

“…The former condition ensures that v * m (ω)v n (ω) is integrable in [0, ∞) and lim ω→∞ v * m (ω)v n (ω) = 0, while the latter condition implies that the rational function v * m (ω)v n (ω) = O(ω) as ω → +0. The form factors with such properties are often found in actual systems involving the process of the spontaneous emission of photons from the hydrogen atom, 15,22 and quantum dots. 16 We do not treat the algebraic form factor that behaves as O(ω 1/2 ) as ω → +0 instead, associated with the photodetachment of electrons from the negative ion 8,12,13,25 and the spontaneous emission from the atoms in the photonic crystals;…”

“…In order to make the condition (50) be satisfied, at least Γ 1 should not vanish identically. We can find such form factors in the physical systems for the spontaneous emission process of photons from the Hydrogen atom 15,22 and the quantum dot. 16 On the other hand, the discussion mentioned here immediately implies the fact that if |ψ ∈ M 2 exists, this time it should satisfy…”

“…The Friedrichs model 9,10 describes the system of the finite discrete levels coupled with the continuous spectrum, in which the former can be interpreted as the unstable excited levels of atoms and the latter as the environmental electromagnetic fields. 11,15,16 We emphasize that the model is not restricted to the single level case 7,8,9,10,11,12,13,15,16 but, rather, the N -level case, 10,17,18,19,20,21 In addition, we assume that the square modulus of the form factors vanishes at zero energy with an integer power, 15,16,22 however it is treated without restriction to a specific form to some extent. Furthermore, since we only consider the initial state spanned by the discrete states, it is sufficient for us to see the reduced resolventR(z) that is just the restriction of the resolvent to the subspace spanned by the discrete states.…”

Initial states and the various long-time-behaviors of the unstable multilevel systems

“…We here choose only three excited levels: the 2p state, 3p state, and 4p state. Then, the form factors corresponding to the 2p − 1s, 3p − 1s, and 4p − 1s transitions become as follows [28,29],…”