Absence of elliptic points

In this section we provide an analjdical proof for the absence of regular period-1 islands in the phase space of the positively kicked onedimensional hydrogen atom. This proof may also serve as a template to prove the absence of elliptic points for other important atomic physics systems such as the stretched hehum atom discussed in Chapter 10. 

As discussed in Section 8.1 the phase-space dynamics of the kicked hydrogen atom is governed by the four-step mapping T = T4 o T3 o T2 o Ti, where the partial mappings Tj, j = 1,. ..,4 are defined in (8.1.13) -(8.1.16), respectively. A period-1 point (0,y) of T satisfies T 0,y) = 0,y). This point is unstable if the trace of the Jacobian of T is larger than 2. Since T consists of four steps, the Jacobian J of T is the product of the four Jacobians Jj, j = 1. 4 defined by 

The Jacobian J2 is trivial. We have J2 = 1. The Jacobian J3 can be expressed as the inverse of J according to J3 = where is evaluated at the primed coordinates. Thus, we need to calculate only J and J4. They are given by 

At this point our restriction to the positively kicked hydrogen atom enters. If we confine our attention to 0 only, we can easily obtain a 

This estimate is independent of a particular location in phase space. This implies that all points of phase space, in particular the period-1 points, are linearly unstable. Thus we have proved that the positively kicked hydrogen atom does indeed not possess any first order eUiptic islands in phase space. It is possible to extend this proof to period-iV points and to show that all period-N points, N integer, are Unearly unstable (Bliimel (1993c)). This implies that the positively kicked hydrogen atom is completely chaotic. We emphasize that, as far as we know, the kicked hydrogen atom is the only model for a physically realizable system where a numerically motivated chaos conjecture was followed up by an analytical proof. In this sense the kicked hydrogen atom is a most remarkable system. 

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