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The problem is to calculate the propagation operator

with the time order operator
which solves the Schroedinger
equation. If
is time dependent, one can extrapololate
by dividing the considered time period into smaller sections over
which
is assumed to be constant. Then the propagation
operator assumes the form
with different
dependent on . This
extrapolation makes use of the relation
A function *f* applied to an operator
can
be spectrally decomposed to give

where
are the eigenvalues and
are the projection operators to the eigenspaces with
If the eigenvalues are known, these projection operators can be formed
by successively applying operators of the form
,
where
denotes unity:
with the normalization factor
However, solving the eigenvalue problem is computationally prohibitively
expensive
and the number
of eigenvalues may be infinite, so approximate methods have to be
found to confine the problem to finding a small subset of the set
of eigenvalues. Applying the above formulae directly leads to the
Lagrangian interpolation of the operator (as the operators
are evaluated recursively) which has numerical disadvantages because
adding another sampling point results in the necessity of calculating
a completely new interpolation polynomial.

**Subsections**

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** Up:** Propagation Methods
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Andreas Markmann
2003-10-22