The Lanczos recursion scheme

The Lanczos method is a good example for the non-uniform approach. The operator a function of which has to be calculated is first represented as a tridiagonal matrix on the finite cyclic subspace spanned by the vectors $\left\vert \widehat{O}^{k}\psi _{0}\right\rangle ,\, k=0,\ldots ,n-1$, the Krylov vectors. The vector $\left\vert \psi _{0}\right\rangle$ is called the initial guess because (together with the dimension $n$) it uniquely defines this subspace. Usually $\psi _{0}$ is taken to be the initial state $\psi (t=0)$.

The build-up of the basis vectors for the finite dimensional representation is initialised by

$$\widehat{O}\psi _{0}=\alpha _{0}\psi _{0}\dot{+}\psi _{1}'$$

where $\dot{+}$ denotes the addition of two linearly independent vectors. The next step is

$$\widehat{O}\psi _{1}'=\beta _{0}'\psi _{0}\dot{+}\alpha _{1}'\psi _{1}'\dot{+}\psi _{2}''$$

This equation can be divided by $\sqrt{\beta _{0}'}$, setting

$$\beta _{0}=\sqrt{\beta _{0}'},\, \alpha _{1}=\alpha _{1}',\, \psi... ...si _{1}'}{\sqrt{\beta _{0}'}},\psi _{2}'=\frac{\psi _{2}''}{\sqrt{\beta _{0}'}}$$

to make the matrix representation in the basis $\psi _{k}$ symmetric and go on with the algorithm

$$\widehat{O}\psi _{k}'=\beta _{k-1}'\psi _{k-1}\dot{+}\alpha _{k}'\psi _{k}'\dot{+}\psi _{k+1}''$$

until the norm of the vector $\psi _{k}$ becomes sufficiently small. The matrix representation of $\widehat{O}$ then will be

$$\widehat{O}\left( \begin{array}{c} \psi _{0}\\ \vdots \\ \psi... ...ft( \begin{array}{c} \psi _{0}\\ \vdots \\ \psi _{n-1} \end{array}\right) .$$

This tridiagonal matrix can be numerically diagonalised to the matrix $D$ by a transformation matrix $Z$. The desired propagated wavefunction $\psi (t)$ can then be found by

$$\psi (t)=Z^{\dagger }e^{-\frac{i}{\hbar }Dt}Z\left( \begin{array}... ...s \\ \left\langle \psi _{n-1}\vert \psi (0)\right\rangle \end{array}\right)$$

i.e. if $\psi _{0}=\psi (0)$ only the first column of the resulting matrix is needed.

In the one-dimensional testcase that I have implemented the Lanczos method was the slowest of all mentioned methods. Nevertheless, for a multidimensional system with hundreds of thousands of gridpoints it is still capable of reducing the problem to a much smaller number of basis vectors, so I expect that the efficiency compared to the other methods will be better in these cases. It is also possible to include some knowledge into the method via the initial guess $\psi _{0}$ and thus make it semi-empirical to get a matrix representation of $\widehat{O}$ (or $\widehat{H}$, respectively) of smaller dimension (e.g. LCAO).