The operator to approximate is with the maximal and minimal eigenvectors , , .

This means the factor is a phase factor, and hence

with the Bessel function of first kind of order (the proof for this expression is to be found in in appendix 4.2).typeset@protect @@footnote SF@gobble@opt These formulas were occasionally faulty in some of the papers I am citing.

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

and the coefficients