Derivation and quantisation of Solov'ev's constant for the diamagnetic Kepler motion

The authors report that the hyperbolic form of the Runge-Lenz vector A, i.e. Lambda (A)=4A 2 -5A z 2 , which was shown to be an approximate constant of the diamagnetic Kepler motion by Solov'ev (1982), can be deduced as the lowest non-trivial Birkhoff-Gustavson normal form for a resonant set of generally four oscillators subject to a constraint. Its special case p phi (=m)=0 (i.e. the case of vanishing magnetic quantum number) is shown to agree with the result of Robnik and Schrufer (1985) derived from two oscillators. A systematic scheme of the semiclassical quantisation (the torus quantisation) is discussed on the explicit construction of the genus-one topology, by means of which all the possible, equivalent quantisation formulae are deduced.