Chapter 6: The scaling method of Vergini and Saraceno

Here I explain and analyse the powerful scaling method recently invented by Vergini and Saraceno (VS) [195,194]. It allows a much more rapid solution of the Helmholtz eigenvalue problem (5.5) in a hard-walled billiard (with Dirichlet boundary conditions) than the sweep methods of the previous chapter, or indeed than any other known method. The success of the method is essentially due to a single key relation, namely that eigenstates are quasi-orthogonal on the boundary (or equivalently, a certain tension matrix is quasi-diagonal near the diagonal in a basis of rescaled eigenstates). However, an adequate explanation for the quasi-orthogonality has been sorely lacking. The knowledge gained about the band profile of special deformations in Chapter 3 will provide, for the first time, an understanding of quasi-orthogonality. This improves upon an (incorrect) guess made by VS, and allows an accurate semiclassical estimate to be made of the deviation from exact orthogonality.

I give a complete, presentation of the scaling method (Section 6.1), discuss the utility of higher-order corrections to the basic method (Section 6.2), and analyse the various sources of error (Section 6.3). Conclusions and open issues are brought up in Section 6.5. Appendix I shows how to evaluate higher-order terms in the tension matrix, correcting an oversight (to do with boundary curvature) in the original formulation of VS.

Generally for illustration I will use results for the 2D stadium billiard, the same shape studied by VS. This is simply for the reason that excellent evanescent basis sets are known for this shape, allowing the errors of the method to be distinguished from those of the basis set. In the last section I will demonstrate an application to computing the eigenstates of a stadium deformed by various finite amounts. The overlaps of these states with the undeformed states will be studied. These calculations were published in our work [48]. Note that it is probably useful to have read the previous chapter to fully appreciate this one.