Asymptotics of Rydberg States for the Hydrogen Atom

Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

“…Related heuristic arguments can be found in [28,7]. In the case q = 0 this equivalence is closely related to the Segal-Bargmann transform in appropriate holomorphic spaces which, in one form or another, plays an important role in the semiclassical analysis performed in [34,36,33,35]. Let us comment in more detail on this relation.…”

“…In our analysis generalized coherent states and associated anti-Wick ΨDOs closely related to the Bargmann representation and the Segal-Bargmann transform appear again in a natural way (see Section 2), although their role is different from the one of their counterparts in [33,36,35]. Although this paper is inspired by [37,33,36,35], much of our construction (see Section 4) is based on the analysis of [22]. In [22] it was proven that for V ∈ C ∞ 0 (R 2 ) the trace in the l.h.s.…”

Semiclassical Szegö Limit of Eigenvalue Clusters for the Hydrogen Atom Zeeman Hamiltonian

“…Next, in section 5, we take the semiclassical limit N → ∞ of this last trace by using the Stone-Weierstrass Theorem, the coherent states for the hydrogen atom introduced in [18], and the stationary phase method in order to estimate the expected value of − B 2 hL 3 m , m ∈ N * , between coherent states. We use decay properties of coherent states shown in [18] but, in addition, we need to estimate decay of their derivatives. Finally, an alternate proof of Theorem 1.3 is presented in section 7.…”

“…It can be shown that for α ∈ A given, the state D (N +1) 2 Ψ α,N is highly concentrated along the classical orbit in configuration space associated to α by the inverse of the Moser map and the classical flow of the Kepler problem on the energy surface Σ(−1/2). See references [18] and [20] for details. Let us denote byẼ N,j , j = 1, .…”

“…(142) Equation (142) expresses the fact that the system of coherent states {Ψ α,N } α∈A gives a resolution of the identity for the eigenspace E N . In reference [18], it is shown that the coherent state Ψ α,N concentrates on the Kepler orbit determined by α as N → ∞ in the case when such an orbit is an ellipse.…”