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####

Example 2: Relaxation method.

The operator
to approximate is
with the maximal and minimal
eigenvectors
,
.
This is often referred to as propagation in imaginary time as it results
from the above propagation operator by setting

(Note that the propagation is actually in *negative* imaginary
time.)
If the operator is applied to a mixed state and the result renormalised,
the state with the lowest energy in the mixed state is produced and
all other states are filtered out at the rate
,
where and are the lowest eigenvalues. If
a higher state is sought, the ground state has to be first produced
and in the second run of the method projected out before every renormalization
step. There are operators
an even number, which produce the eigenstate closest to the energy
directly without the need of producing the lower states first
but I shall not go into any details about these [8].

The factor
for the relaxation operator
is *not* a phase
factor anymore,
and hence

with the *modified* Bessel function of first kind
of order n.
This time, the coefficients go to zero exponentially as becomes
greater than
so the method is very fast *except*
for very short times where this exponentially decaying tail becomes
dominant in the calculation time.

The same result is produced by

with the recursion

(note the changed sign of
compared to the second
algorithm in Ex. 1) and the coefficients
This can be shown by proving inductively that
,
as obviously
:

** Next:** The Lanczos recursion scheme
** Up:** The Chebyshev method
** Previous:** Example 1: Propagation operator.
Andreas Markmann
2003-10-22