B) `Green's function matching' (sweep in energy)

Class B can be applied when the Green's function (i.e. the eigenstates of $\hat H$ the in $\mathcal{D}$ with some different BCs) is known analytically. It has the advantage that this Green's function can now be used to propagate between points on $\Gamma$ without explicit solution over the volume of $\mathcal{D}$. This reduces the dimensionality of the problem from $d$ to $d{-}1$. Why is this an enormous reduction in computational effort? Generally the number of bases $N$ is reduced from $O({\mathsf{V}}/\lambda_{{\mbox{\tiny B}}}^d)$ to $O({\mathsf{A}}/\lambda_{{\mbox{\tiny B}}}^{d-1})$, where ${\mathsf{V}}$ is the volume of $\mathcal{D}$ and ${\mathsf{A}}$ is the area of $\Gamma$. The typical quantum wavelength at the energy of interest is $\lambda_{{\mbox{\tiny B}}}$. This is a typical reduction in $N$ of $O({\mathsf{L}}/\lambda_{{\mbox{\tiny B}}})$ where ${\mathsf{L}}$ is the typical length scale of the region. Given that the required dense matrix techniques take time $O(N^3)$, this is a huge reduction in effort when there are many wavelengths across the system (semiclassical limit). This corresponds to very high quantum number of interest $n_{{\mbox{\tiny E}}}$. The problem has now become a one of matching on the boundary. If the length in coefficient space $\vert{\mathbf x}\vert^2$ roughly corresponds to a state's norm in $\mathcal{D}$, then the BC matching is equivalent to a zero-determinant condition (see Table 5.1). The $E$ values for which the BCs can be obeyed must be found. However the linearity in $E$ of the matrix equation has been sacrificed. There is usually no way to tell at what $E$ the determinant will vanish, so a `sweep' or `hunt' (search routine) in $E$-space is required, taking many iterations per solution found.

The Boundary Integral Method (BIM or BEM), probably the most common example of Class B methods, has been used extensively in engineering and physics [36,121,28,33,34]. It emerged in the engineering community in the 1950s (see Chapter 1 of [36]). Also in Class B are similar methods which replace matching on $\Gamma$ by `internal boundary conditions' where two (or more) different Hamiltonians may be joined (matched) along an internal surface, which is simply a quantum version of a Poincaré surface of section. Examples are Bogomolny's method [34] (where the propagation in each half-region is found semiclassically), the variation of Prosen [163] (where the Hamiltonians are known analytically), the variation of Smilansky [172], and the Korringa-Kohn-Rostoker (KKR) method (as first used in the quantum chaos community by Berry [23], also see [162] and references within). Equally important for quantum billiard problems is Heller's PWDM, upon which this chapter is based, discussed below. The above methods differ mainly in the choice of basis set, and the way in which the matching is expressed. However they all suffer from the need to `sweep' along the energy axis.

For specialized problems other Class B methods have been invented, for instance the t-matrix inversion technique of Lupu-Sax [140] for eigenstates of domains containing point-scatterers. Note that the standard `shooting method' [119] for problems in $d=1$ (where it is possible to integrate directly the ordinary differential equation resulting from the position-space form of (5.1)) is essentially a Class B method because an energy sweep is required.