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## Computation of overlap matrix and its profile

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 () inside ( ). Then , where is the boundary of the intersection region . Then the overlap integral over can be written

 (6.30)

In practice this formula is evaluated using the method of Appendix H, using the wavefunctions evaluated at the set of discretization points from and which lie in . The formula becomes inaccurate (depending on the accuracy of the eigenstates) when . In practice this was only a problem at wavenumber differences well below the mean level spacing. The above overlap calculation technique is so efficient that it only took a few seconds to fill each matrix of size about by , at .

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 and 6.15. Comparison with the first-order perturbation theory (FOPT) result,

 (6.31)

is also made in these plots. is explicitly given by (C.2), and its band profile is described in Chapters 2 and 3. The FOPT result simply corresponds to a linearization of the wavefunction using the notation of Appendix I. The profile shows complete breakdown of FOPT when reaches , and in the case of a generic deformation the formation of a `core' region before this. For more analysis see our work [48].

Next: Discussion Up: Application: local density of Previous: Deformations and eigenstate computation
Alex Barnett 2001-10-03