For we have for each component
of the dipole (position) operator,

where as before . This follows from row 3 of the matrix . Summation over is implied. A mixture of vector and Einstein notations has been avoided. In the Dirichlet BC case this becomes

Note that the integral is proportional to the matrix element of the billiard deformation corresponding to translation in the direction.

No form for has been found. The corresponding unit vector does not lie in the row space of , indicating that other boundary derivatives are needed as input.

Alex Barnett 2001-10-03