When a billiard is deformed, the potential
changes
by an infinite amount (in the hard-wall limit)
in narrow regions of space on the boundary.
These regions are simply those excluded from or newly-included into the
cavity interior by the deformation.
The operator
is therefore either infinite or zero in a position
representation.
Therefore it might be a surprise that the
matrix elements
in the energy representation are in fact
finite.
The following derivation has been known for some time
[28,118,46].
The position of a particle in the vicinity of a wall element
is conveniently described by
,
where
is a
dimensional surface coordinate
and
is a perpendicular `radial' coordinate.
I set
outside the undeformed billiard; this will relax
the usual Dirichlet condition
on the boundary (at
).
Later I will take the limit
.
I have
Therefore, given a subset of adjacent eigenstates, an on-diagonal
block of
the matrix
can be found.
A resulting example is shown in Fig. 2.7.
The integrals are evaluated using the techniques of Appendix G.