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Numerics.TDSE.Chebychev

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General scheme

The basic idea of the Chebychev method is an expansion into Chebychev polynomials. To recall, the idea of a (truncated) expansion in an orthonormal basis of functions is to write a given function f(x) as

$f(x) = \sum_{n=0}^N f_n \varphi_n(x)$

For a given metric induced by a scalar product, the expansion is optimal if the coefficients are chosen as

$f_n = \langle \varphi_n | f \rangle$

Chebychev expansion

For the particular problem described in the section on propagators in quantum dynamics, we choose an expansion of the form

$e^{-i\alpha x} = \sum_{n=0}^N c_n(\alpha) T_n(x)$

Here, x is an arbitrary eigenvalue of the normalized Hamiltonian in the range from -1 to +1, and we choose to consider α as a scaling factor that determines the coefficients, while T_n are Chebychev polynomials of the first kind. These are orthogonal, but not normalized with respect to the scalar product

$\langle T_n | T_m \rangle = \int_{-1}^1 T_n(x) T_m(x) \frac{dx}{\sqrt{1-x^2}} = \frac{\pi}{K_n} \delta_{nm}$

Here, K=1 for n=0 and K=2 otherwise. The coefficients are then calculated according to

$c_n(\alpha) = \frac{K_n}{\pi} \langle T_n | e^{-i\alpha x} \rangle = \frac{K_n}{\pi} \int_{-1}^1 \frac{e^{-i\alpha x} T_n(x)}{\sqrt{1-x^2}} dx$

(the factors come from the missing normalization). Question: How do we evaluate the integral? For this, we use an integral representation for the Bessel functions of first kind (see Abramowitz&Stegun alias DLMF, eq. 10.9.2),

$J_n(z) = \frac{i^{-n}}{\pi} \int_0^\pi e^{iz\cos\theta} \cos(n\theta) d\theta$

We can transform the integral using two steps:

We use the property of Chebychev polynomials $T_n(\cos\theta) = \cos(n\theta)$ (see Abramowitz&Stegun alias DLMF, eq. 18.5.1),
We substitute the integration variable to x=cosθ

and obtain the result

$T_n(\cos\theta) = \cos(n\theta)$

which looks almost like what we want for z=-α. Note that from the definition and the real-valuedness of the Bessel function, it is easy to show that Bessel functions are even (/odd) for even (/odd) n. We end up with an expression for the coefficients that is

$c_n(\alpha) = i^n (-1)^n K_n J_n(\alpha)$

Putting all this together, and replacing x by the normalized Hamiltonian, we end up with the final form of the Chebychev propagator

$e^{-i \alpha \hat H_\text{norm}} = \sum_{n=0}^N i^n (-1)^n K_n J_n(\alpha) \ T_n(\hat H_\text{norm})$

If you look up the original references by Tal-Ezer and Ronny Kosloff, you will find that their expansion looks considerably different. There, some of the pre-factors are included in the polynomials, thus defining new polynomials as

$\phi_n(x) = i^n (-1)^n T_n(x)$

If you plug in these definitions into the recursive definition of the Chebychev polynomials, you get the recursion relations for the new polynomials φ_n that are shown on the page for solution methods for the time-dependent Schrödinger equation.

Wiki: Numerics.TDSE