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
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,
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(6.31) |