Use of a scaling basis

Because the scaling eigenfunctions $\psi_\mu(k,{\mathbf r})$ defined above all have the same $k$, and are normalizable, they fall into the above linear space. They can be represented by coefficient vectors ${\mathbf x}^{(\mu)}$ of a Helmholtz basis set with a single wavenumber $k$,

$$\psi_\mu(k,{\mathbf r}) \; = \; \sum_{i=1}^N x_i^{(\mu)} \phi_i(k;{\mathbf r}) + \varepsilon _\mu(k;{\mathbf r}) ,$$ (6.17)

where $\varepsilon _\mu$ is some small error function for each state. (The error function is a Helmholtz solution and is orthogonal to the basis set space). The functions $\phi_i$ will not be orthogonal over $\mathcal{D}$, and there will be some (numerical) null-space of vectors ${\mathbf x}$ that has negligible effect on wavefunctions inside $\mathcal{D}$ (see Section 5.3.1).

The boundary tension $f(k)$ is a quadratic form in the linear space. Applying this form to the error $\varepsilon _\mu(k;{\mathbf k})$ gives the achievable tension minimum $\epsilon_\mu$ for each state. In the $N$-dimensional above basis, the quadratic form is written ${\mathbf x}^{{\mbox{\tiny T}}} F(k) {\mathbf x}$ where $F$ is the tension matrix.

In the methods of the previous chapter the only requirement on the $k$-dependence of this basis set was that at a given $k$ all the functions are Helmholtz solutions at that $k$. In this chapter we must specialize to a scaling basis,

$$\phi_i(k;{\mathbf r}) \; = \; \phi_i(k,{\mathbf r}) \; \equiv \; \bar{\phi}_i(k{\mathbf r}) \hspace{0.4in} \forall i .$$ (6.18)

Note that $(k,{\mathbf r})$ implies a scaling function whereas $(k;{\mathbf r})$ implies some more general $k$-dependence. The expansion of any scaling function, as in (6.17), now has a coefficient vector ${\mathbf x}$ which is constant as $k$ changes. This will be necessary for our particular case of the scaling eigenfunctions. Note that certain basis sets, most notably that of the BIM, cannot be used because their basis functions (Green's functions) have origins at different points `pinned' to the boundary $\Gamma$.