By Primack, Smilansky.
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Extra info for Quantum 3D Sinai billiard: a semiclassical analysis
Example text
We "nally have to show that there exists only a single 5 minimum of ¸ K . The complication here is that, in principle, a minimum of ¸ K does not necessarily 5 5 correspond to a minimum of ¸ I , since there are, in general, more variables in the latter. We resolve 5 this di$culty by using arguments from the proof of Schanz [32] as follows. A necessary condition for minimality is that orbits are either externally re#ected from the scatterers or cut through them in straight segments. Internal re#ections are not allowed for a minimum.
Since the 3D Sinai is meant to be a paradigm for 3D systems, we must remove the in#uence of the non-generic bouncing-ball families and "nd a way to focus on the contributions of the generic and unstable periodic orbit. This is imperative, because in the 3D Sinai billiard the 54 H. Primack, U. Smilansky / Physics Reports 327 (2000) 1}107 Fig. 28. 2 that contains only generic, unstable periodic orbits. In all cases k"160, "30. The locations of the bouncing balls are indicated: daggers for 2-parameter bouncing balls that occupy 3D volume in con"guration space, stars for 2D bouncing balls and crosses for 1D bouncing balls.
Smilansky / Physics Reports 327 (2000) 1}107 39 Fig. 22. The function P(l) (cf. RHS of Eq. 3 and "tted according to Eq. (69). We also show the asymptotic prediction (66). The sum-rule (66) which formed the basis of the previous analysis is an expression of the ergodic nature of the billiards dynamics. , taking the surface of the sphere and the tangent velocity vector as the PoincareH section. The resulting return-map excludes the bouncing-ball manifolds since they do not intersect the section. ect is noticed because between successive collisions with the sphere the trajectory may re#ect o!
Quantum 3D Sinai billiard: a semiclassical analysis by Primack, Smilansky.
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