Predrag,

I since found the paper I was referring to. It is this one:

www.nbi.dk/~predrag/papers/tunnel.ps.gz

The problem which Stephen mentions is interesting. We looked at that problem because the splitting statistics in the nonsymmetric case are, in some sense, "generic" whereas the axially symmetric problem has non-generic statistics. Since that tied into random matrix theory view of the world, we considered the broken axial symmetry as a numeric example. The axially symmetric problem has interesting non-generic statistics arising from a careful consideration of the complexified classical dynamics. Presumably it is a qualitatitively different picture when the axial symmetry is broken since the nature of the fluctuations around the mean are very different.

A related point is that, as far as I am aware, noone has ever shown that tunnelling splittings from an inversion symmetry are necessarily positive. We understand this on a semiclassical level since they are the expectations of a positive-definite operator. But I am not aware of any fully quantum proof of that fact in more than one dimension. In one dimension there are interleaving theorems which guarantee it. Part of the problem is defining in a fully quantum view of the world what you mean by the term "splittings". The identification of doublets is an inherently semiclassical notion although, again, it can be given a precise meaning in one dimension. One way might be to prove that the first time that two even states occur sequentially in the spectrum is at energies above the classical energy saddle. Or in an hbar quantisation, it would be enough to prove that the even and odd states interleave if the energy is below the saddle. This is probably not a problem which Olivier will solve in a month but it is an interesting one to think about for someone to solve. [Assuming it hasn't already and I just don't know about it.]

I would like to have that.If someone needs this I have the PRE reprint in pdf.

Niall