Representation of exact and semiclassical eigenfunctions via coherent states. Hydrogen atom in a magnetic field

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

“…Following [9], we regularize the Hamiltonian (1). For each n ≥ 1 and E < 0, we introduce parameters ν and μ, a new variable q ∈ R 3 , and a functionψ by the formulas (3) E = − 1 4n 2 ν 2 , μ = ε 2 n 6 ν 4 , q = x n 2 ν , ψ(x) =ψ (q) n 2 .…”

“…Then μ 1 and we can apply a quantum version of the averaging method [19], [20], [9] to the problem (4). The basic idea of this method is to find an invertible operator U and an operator S 1 + μS 2 such that…”

“…Since in our problem the spectrum of the operator S 0 is integral, the condition (10) is satisfied. The operator U can be explicitly computed [9]. Define…”

“…An algebraic method for constructing the asymptotics of the spectrum and the eigenfunctions of the operator (1) was introduced in [9]. It is based on an algebraic averaging of the perturbation, followed by the passage to the symmetry algebra, and a coherent transformation from the original representation of this algebra to an irreducible representation of that algebra in the space of functions over a Lagrangian submanifold in a symplectic leaf.…”

“…A possible approach to constructing asymptotics near cluster boundaries using "deformed" coherent states was outlined in [9], but so far it has not been justified in higher approximations.…”