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We now proceed term-by-term in (I.2). The zeroth-order
is
![\begin{displaymath}
\psi \ \stackrel{{\mbox{\tiny BCs}}}{\longrightarrow}\ 0,
\end{displaymath}](img1920.gif) |
(I.15) |
where the arrow implies application of the Dirichlet boundary conditions (BCs),
namely
for all
(remember that we already chose
in order to be able to write the metric).
The first order is
![\begin{displaymath}
{\mathbf a}\cdot\nabla \psi \ = \ a_n\psi^n + a_t\psi^t \ \stackrel{{\mbox{\tiny BCs}}}{\longrightarrow}\
a_n\psi^n ,
\end{displaymath}](img1922.gif) |
(I.16) |
where the result
was used. In will use in future the fact that all
higher
-derivatives
vanish upon application of the BCs.
The second order requires application of the rules (I.10):
The Helmholtz equation and the BCs together imply
, a relation used both above and below.
In general our task is to reduce the number of
-derivatives; it is always
possible using such manipulations to leave only terms containing a single
-derivative.
This will be desirable for manipulation of boundary integrals by parts later.
The third-order result (included below) requires use of the following.
simplifies to
when the BCs are applied.
This required the normal derivative of curvature
since
now needs to be regarded as a scalar field with
-dependence.
The tangential derivative is given the name
.
Also
simplifies to
when BCs are applied.
Combining everything and finally substituting for
gives
the expansion of a Dirichlet scaling eigenfunction at location
on
:
The expression is believed to be correct to order
. The
complexity increases greatly with each power of
, and higher derivatives
of the curvature
enter.
However, in the case of the circle billiard
(where
is constant) it is easy to verify the expansion.
One general pattern is that the
-order terms all contain
powers
of
, and for each term the number of spatial derivatives of
must equal the power of
minus the power of
.
It is important to notice
that the
term differs by the presence of a factor of
from the
term given by VS.
Next: Tension matrix expansion
Up: Appendix I: Scaling expansion
Previous: Curvilinear boundary coordinates
Alex Barnett
2001-10-03