The basic scaling method

We are interested in finding eigenstates of a $d$-dimensional billiard whose domain $\mathcal{D}$ has `volume' ${\mathsf{V}}$ and whose boundary $\Gamma$ has `surface area' ${\mathsf{A}}$. Therefore the typical system size is ${\mathsf{L}}\sim {\mathsf{V}}/{\mathsf{A}}$. The scaling method will find all the states in a range $O({\mathsf{L}}^{-1})$ around a given wavenumber $k$, using a single diagonalization of the same numerical effort required for each $k$ evaluation in the previous chapter. Up to $N_{sc}/10$ states of useful accuracy are returned per diagonalization, making the method many orders of magnitude faster than any other method known at this time.