Classical ladder functions for Rosen–Morse and curved Kepler–Coulomb systems

Abstract

Ladder functions in classical mechanics are defined in a similar way as ladder operators in the context of quantum mechanics. In the present paper, we develop a new method for obtaining ladder functions of one dimensional systems by means of a product of two 'factor functions'. We apply this method to the curved Kepler-Coulomb and Rosen-Morse II systems whose ladder functions were not found yet. The ladder functions here obtained are applied to get the motion of the system.

Superintegrability on the three-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the sphere S 3 and on the hyperbolic space H 3

“…(ii) The study of classical superintegrability can also be considered as a first step for the study of the corresponding quantum versions (we recall that quantum superintegrability is related with the degeneracy of the energy levels as in the hydrogen atom). The behaviour of the functions M jκ and N jκ shows a certain relation with the properties of classical ladder functions studied in [71]. An interesting point is if the quantization of the functions M jκ and N jκ as appropriate operators can be related with quantum ladder operators.…”

Ladder operators and coherent states for the Rosen-Morse system and its rational extensions

“…This case arises when u(x) is the ground state ψ(0; x) of H with ε = E(0). Every state of H is connected to one state of H according to (15), except for the ground state since B − ψ(0; x) = 0. The energy level E(0) is removed of the spectrum of H during the transformation: Sp( H) = Sp(H) \ {E(0)}.…”

“…This case arises for unbounded seed solutions u(x) with ε < E(0) such that 1/u(x) is normalizable. All the states of H are connected to that of H according to (15). It is known however in this case that B + annihilates 1/u(x), making it a normalizable eigenstate of H with energy ε < E(0).…”

“…The well is dug from the RMII system to the rational extension in order to allow the additional energy level ε. The normalizable eigenstates for the rational extensions are obtained via the correspondence (15) for state-adding SUSY:…”

“…This system admits a finite discrete bounded spectrum and has been studied in different contexts [8,9,10,11,12,13]. Recently, a realization of the ladder operators was obtained for this system through an analogy with classical mechanics [14,15]. In this paper, we intend to motivate a realization of the ladder operators for the RMII system from a purely quantum mechanical point of view.…”