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Discrete Variable Representation (DVR) [1], [5]
A wavefunction is represented as a linear combination
of given interpolation functions by

(1) 
such that for given interpolation points
If the satisfy the orthonormality relationin the arguments

(2) 
then this interpolation expression, i.e. the matrix
can be directly inverted to give

(3) 
for then
If the
furthermore satisfy the orthonormality
relation in their order

(4) 
or, more generally,
then, choosing the sampling points as the zeros of the
Nth polynomial , it follows from Gaussian integration
theory that their discrete representations also fulfill the orthonormality
relation

(5) 
with constant point weights because their degrees are
all smaller than (see appendix 4.1). The coefficients
become
In this case, the scalar product formula simplifies to
with the coefficients of the representation of .
Substitution of the coefficients (3) in the representation
(1) results in
with orthonormal coordinate eigenfunctions
and the sampling values :
Then the derivative operators can be written as
Next: Comparison
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Andreas Markmann
20031022