Next: Curvilinear boundary coordinates
Up: Appendix I: Scaling expansion
Previous: Appendix I: Scaling expansion
Expansion of the dilation operator
A scaling wavefunction (see Eq.(6.1)) whose
unscaled wavenumber is will be rescaled to a new wavenumber
, where is the wavenumber shift.
Its value at location is

(I.1) 
where the translation vector is
.
Formally this translation can be performed by exponentiation of the
generator of translation [174,7]:

(I.2) 
Thus we have written the scaled value in terms of local derivatives of the
unscaled wavefunction.
This becomes a definition of dilation only when the dependence
of is finally substituted.
Note that is held constant as far as is concerned.
(This differs from other possible definitions of dilation [14],
where the commutator
plays a role;
however my form
will be simpler to expand to high order).
For instance in Cartesian coordinates,
is to be interpreted as
(Einstein summation
assumed), because in Cartesian coordinates
.
Figure I.1:
Curvilinear coordinates used to represent points near the
billiard boundary .
The orthogonal unit vectors are the vector fields and .

Next: Curvilinear boundary coordinates
Up: Appendix I: Scaling expansion
Previous: Appendix I: Scaling expansion
Alex Barnett
20011003