The operator is expressed as an expansion in Chebyshev polynomials

with the coefficients

This inversion formula is using the orthonormality relation of type (4)

In order to use the Chebyshev polynomials, the range of eigenvalues of the operator has to be adjusted by the substitution

The summation of the derived vectors is continued until the deviation of the coefficients from zero drops below a given accuracy. This makes it possible to achieve any desired accuracy up to computer accuracy.

The Chebyshev method is preferentially used for propagation of time-independent
operators for otherwise it has to be run several times for subintervals
of time over which it is *assumed to be constant*. The same has
to be done if intermediate states in time are sought. Both will reduce
the efficiency of the method.