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,

$$(\nabla^2 + k_a^2) a{({\mathbf r})}\; = \; 0 ,$$ (H.1)

where the Helmholtz solution $a{({\mathbf r})}$ is in general a complex-valued function of space. At a given $k$, Green's theorem tell us that a Helmholtz solution in a region $\mathcal{D}$ is determined uniquely by its value or normal derivative on the enclosing surface $\Gamma$. Therefore domain integrals of the type

$$\left\langle a\right\vert\hat{O}\left\vert b\right\rangle _{... ...! d{\mathbf r} \, a^*{({\mathbf r})}(\hat{O} b){({\mathbf r})}$$ (H.2)

can be reduced to integrals of local quantities over $\Gamma$ alone, when $a$ and $b$ are Helmholtz solutions. This is true whether the corresponding wavenumbers $k_a$ and $k_b$ 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 $k{\mathsf{L}}$ is expected for simple geometries, for a typical system size ${\mathsf{L}}$. 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 $a$ and $b$ are constant-energy solutions, (H.2) can be called a `matrix element' of the operator $\hat O$ in the energy basis.

I collect together useful results for such evaluations, in the cases $\hat{O} = 1$ (corresponding to a simple overlap integral), $\hat{O} = {\mathbf r}$ (dipole operator), $\hat{O} = \nabla$ (momentum operator), and $\hat{O} = {\mathbf r} \cdot \nabla$ (differential dilation operator). The results apply to all spatial dimensions $d\ge2$. 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 $\Gamma$, unless I state otherwise. This allows $\Gamma$ 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 $a^*\!{\leftrightarrow}b$ will imply a duplication of the previous term except with $a^*$ and $b$ swapped.

Some of the results have already been known to other authors [28,33], and some are new, especially the case of general $d$. 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.