The Chebyshev method

The operator is expressed as an expansion in Chebyshev polynomials

$$f(z')=\frac{b_{0}}{2}+\sum ^{\infty }_{n=1}b_{n}T_{n}(z')$$ (10)

with the coefficients

$$b_{n}=\frac{2}{\pi }\int\limits ^{1}_{-1}\frac{f(z')T_{n}(z')}{\sqrt{1-(z')^{2}}}dz'.$$ (11)

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

$$\frac{2}{\pi }\int\limits _{-1}^{1}\frac{T_{m}(z)T_{n}(z)}{\sqrt{... ...begin{array}{ll} \delta _{mn} & ,\, m,n>0\\ 2 & ,\, m=n=0 \end{array}\right.$$

of the Chebyshev polynomials - substitute the series expansion (10) of f in (11).

In order to use the Chebyshev polynomials, the range of eigenvalues of the operator $\widehat{O}$ has to be adjusted by the substitution

$$\widehat{O'}=2\frac{\widehat{O}-\lambda _{min}\widehat{I}}{\lambda _{max}-\lambda _{min}}-\widehat{I}$$

which results in the necessity of multiplying a factor $\Phi$ after the calculation to get the result. The images $\phi _{n}=T_{n}\left( \widehat{O'}\right) \psi$ are then built up by the recursion relation for Chebyshev polynomials:

$$\phi _{0} = \psi ,$$

$$\phi _{1} = \widehat{O'}\psi ,$$

$$\phi _{n+1} = 2\widehat{O'}\phi _{n}-\phi _{n-1}.$$

The summation of the derived vectors $b_{n}\phi _{n}$ 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.