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

The weighting is simply the deformation function . Note that for any

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.