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.]
If someone needs this I have the PRE reprint in pdf.