Application: local density of states at finite deformations

Now I present a calculation of eigenstates in a billiard deformed by various finite amounts. Note that the considerations of Chapter 3 only dealt with infinitesimal deformations (rates of change under deformation). We use the same notation as that chapter, namely that the parameter $x$ causes a normal displacement of the billiard wall by $x D{({\mathbf s})}$. The matrix of overlaps of eigenstates at deformation $x$ with the undeformed eigenstates will be computed. From this follows an estimate of the profile of the matrix, taken by averaging the off-diagonal strength. This is otherwise known as the local density of states (or `line shape') of the deformation. The physical significance and analysis of this profile is given in [48].

The billiard used was the 2D quarter stadium (Fig. 2.6), chosen because of the excellent RPW and EPW Helmholtz basis set known for this shape (Section 6.3.3, the previous chapter, and Appendix J).