The quantity of interest is the matrix of overlaps . The eigenstates are dependent on the deformation parameter . The overlap is taken in position-space over the domain , defined as the intersection of the undeformed domain and the deformed domain . We seek a way to perform these -dimensional domain integrals rapidly (in our case ). Appendix H shows that because the eigenvalues and are (in general) different, there exists a very simple formula (H.5) for computing this integral using only the boundary. This will be a huge saving in effort.
Because our eigenstates have Dirichlet boundary conditions, a simplification
can be made. Define () as the part of
, where is the boundary of the
intersection region .
Then the overlap integral over can be written
Given the overlap matrix at each , its average profile was found by the smoothing method of Section C.2. The smoothing width was chosen to be 0.02 in wavenumber units, which was a couple of times the mean level spacing .
The resulting sequences of profiles are shown in Fig. 6.14
Comparison with the first-order perturbation theory (FOPT) result,