Propagation methods [2]

The problem is to calculate the propagation operator

$$\widehat{U}(t_{e},t_{s})=\widehat{T}e^{-\frac{i}{\hbar }\int\limits _{t_{s}}^{t_{e}}\widehat{H}(t')dt'}$$

with the time order operator $\widehat{T}$ which solves the Schroedinger equation. If $\widehat{H}$ is time dependent, one can extrapololate by dividing the considered time period into smaller sections over which $\widehat{H}$ is assumed to be constant. Then the propagation operator assumes the form

$$\widehat{U}(t_{e},t_{s})=e^{-\frac{i}{\hbar }\widehat{H}_{s}(t_{e}-t_{s})}$$

with different $\widehat{H}_{s}$ dependent on $t_{s}$. This extrapolation makes use of the relation $\widehat{U}(t_{1}+t_{2},t_{0})=\widehat{U}(t_{2},t_{1})\widehat{U}(t_{1},t_{0}).$

A function f applied to an operator $\widehat{O}$ can be spectrally decomposed to give

$$f\left( \widehat{O}\right) =\sum _{n}f(\lambda _{n})\widehat{P}_{n},$$

where $\lambda _{n}$ are the eigenvalues and $\widehat{P}_{n}=\left\vert u_{n}\right\rangle \left\langle u_{n}\right\vert$ are the projection operators to the eigenspaces with

$$\widehat{P}_{n}\widehat{P}_{m}=\delta _{mn}\widehat{P}_{n}.$$

If the eigenvalues are known, these projection operators can be formed by successively applying operators of the form $\widehat{Q}_{i}=\left( \widehat{O}-\lambda _{i}\widehat{I}\right)$, where $\widehat{I}$ denotes unity:

$$\widehat{P}_{n}=\frac{1}{\aleph }\widehat{Q}_{N}\cdots \widehat{Q}_{n+1}\widehat{Q}_{n-1}\cdots \widehat{Q}_{1}$$

with the normalization factor

$$\aleph =\left( \lambda _{n}-\lambda _{N}\right) \cdots \left( \la... ...bda _{n}-\lambda _{n-1}\right) \cdots \left( \lambda _{n}-\lambda _{1}\right) .$$

However, solving the eigenvalue problem is computationally prohibitively expensive $\left( O\left( N^{2}\right) \right)$ 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 $\widehat{Q}_{i}$ are evaluated recursively) which has numerical disadvantages because adding another sampling point results in the necessity of calculating a completely new interpolation polynomial.