S Creagh Feb 11, 2004
> > 2. We never looked in detail classically at problems where the central
> > orbit is nonperiodic.
> > How can the central orbit not be periodic? I thought it always was by
> > continuity arguments. Can you give an example where it is not?
> suppose you have x -> -x symmetry, but not y -> -y symmetry. Then the
> shortest tunneling orbit is a little arch, and it shoots off a classical
> orbit not along the x axis, but at some finite angle. With probability
> one, this does not close a perididc orbit, and certainly not a short
> one.
Predrag is correct here. Examples can be found in PRL 84, 4084 (2000)
where Niall and I looked at statistics for such nonsymmetric problems.
As for Simonotti and Saraceno, it might help to look at equation (12)
of their paper. The quantitity g(k) is constructed for billiards but
there are fairly obvious analogs for smooth potentials --- replace k by E
and let T(k)->T(E) be the Bogomolny transfer operator. In any case g(E)
as constructed by Simonotti and Saraceno is such that
g(E_n) = |n> is the chaotic eigenstate. The splitting for this state is
then evaluated semiclassically from
\Delta E_n = = Tr [ Tun(E_n) |n> SOS -> .................... -> SOS -> SOS
ie i applications of a real SOS map F are composed with a complex map W
at the end. These fixed points are precisely the complex periodic orbits but
organised according to the number of bounces i in the "real" segment.
This might in general be SOS dependent but in the potentials we looked at
there's a failry natural way to do it.
[For the coherent states Simonotti and Saraceno look at you get uglier
sums with uglier boundary conditions on the orbits.]
There are also things like c_i(k)~c_i(E) which are computed from real pos.
The denominator also involves nothing but real pos and could be calculated
without reference to tunnelling. It's something you could try to replace
with some qm spectral calculation for a single well if you were
sufficiently uninterested--- this is what I was getting at in my first email
when I mentioned doing a partly quantum calculation.
Anyway (12) essentially gives the splitting \Delta E_n explicitly as a sum
over complex pos, combined with sums over real pos which bring in
nontunnelling spectral information.
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Stephen Creagh
School of Mathematical Sciences
Division of Applied Mathematics phone: (44) 115 951 3853
University of Nottingham fax: (44) 115 951 3837
NG7 2RD, UK.
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