Appendix H: Boundary evaluation of matrix elements of Helmholtz solutions

Here I collect some useful tools for expressing wavefunction
domain integrals purely on the boundary.
The time-independent (constant-energy)
wave equation in uniform space is the Helmholtz
equation,

can be reduced to integrals of local quantities over alone, when and are Helmholtz solutions. This is true whether the corresponding wavenumbers and are equal or unequal. This has a vast numerical advantage in the semiclassical limit: a reduction of the number of function evaluations by a factor of a few times is expected for simple geometries, for a typical system size . Once an expression in terms of surface integrals is known, the technique of Appendix G can be used for the numerical evaluation. Note that since and are constant-energy solutions, (H.2) can be called a `matrix element' of the operator in the energy basis.

I collect together useful results for such evaluations, in the cases (corresponding to a simple overlap integral), (dipole operator), (momentum operator), and (differential dilation operator). The results apply to all spatial dimensions . For each operator the cases of equal and unequal wavenumbers need to be handled separately. I have maintained complete generality of the boundary conditions (BCs) on , unless I state otherwise. This allows to be chosen arbitrarily, in particular, independently of the location of wavefunction nodes, billiard walls, or other features. From these the special cases corresponding to Dirichlet, etc., BCs can be derived--I will present some of these simpler forms below. Throughout this appendix the symbol will imply a duplication of the previous term except with and swapped.

Some of the results have already been known to other authors [28,33], and some are new, especially the case of general . For those that are known, my derivations are much simpler than the complicated vector manipulations used by those authors. The most powerful new techniques I have found to be the use of scaled solutions (see Section 6.1.1), and a matrix inversion trick of M. Haggerty. Both are presented below. Remember that the goal in what follows is to `push' all volume terms onto the boundary.