Examples of a `special' deformation and a `generic' deformation were used (see Chapters 3 and 4). The special deformation chosen was rotation about the origin (lower-left corner). This corresponds to (see Table 3.2). Because the special deformations preserve shape, the eigenstates of a thus-deformed billiard are unchanged (space is invariant under translation and rotation). Therefore only one set of eigenstates is required for all values.
The generic deformation used was , where is the angle on the circular part of the boundary measured from the vertical. on the top, left, and bottom straight sections of the boundary. This has the effect of pushing out the curved part into an ellipse (and a corresponding lengthening of the bottom straight section). This deformation was chosen because it was possible to create large deformations with minimum sacrifice to the quality of the basis-dependent tension minima .
Generally, it has been found that almost any non-special deformation of the stadium creates a shape for which the RPW basis becomes much worse, and that improvement by addition of EPWs is also limited. For deformations that are localized on the perimeter, this renders the problem unsolvable if approaches (de Broglie wavelength) or larger. In the above choice of deformation at it was possible to deform from to with a corresponding change in typical tension from to . A few states of the deformed system had higher tensions of . This was acceptable for the calculation.
At each , all the states in the wavenumber range were gathered using multiple applications of the scaling method at choices of equally spaced by in wavenumber. Thus only states falling within were kept from each scaling diagonalization. This rather small value of means that only about 2 states were found per diagonalization. However, because of the limitation of the basis set in such deformed shapes, this gave the highest accuracy. Because the scaling method is so fast, it did not slow down the overall computation by much (the evaluation of eigenstate values and gradients on the perimeter was the main bottleneck).
To illustrate the effect of deformation on the eigenvalues, Fig. 6.13 shows their parameter-dependence under a (different) generic deformation. This was generated (to lower accuracy) by a single scaling diagonalization at each value.