This material is based on the lectures given by Prof. A. C. Shi from Jan. 12 to 14, 2009 at Fudan University, with some additional derivation details.

## 1. One Mode Approximation

Assume the function to be approximated can be expanded in a series

$\phi (x) = \sum_n \phi_n (x) e^{ik_nx}$

where $n = 0, \pm 1, \pm 2, \cdots$. For one mode approximation, only the first three terms are taken, that is

$\phi (x) = \phi_0 (x) e^{ik_0x} + \phi_{-1} (x) e^{ik_{-1}x} + \phi_1 (x) e^{ik_1x}$

In two and higher dimensional space, variables, $x$ and $k$, become vectors $\mathbf{x}$ and $\mathbf{k}$. To approximate high dimensional functions, one should consider the origin and its neighbors. Using two-dimensional hexagonal lattice as an example, one mode approximation retains only the origin term and the six nearest neighbors, depicted in Figure 1.

One mode approximation is not a good approximation for square wave. To expand square wave function according to eq. 1, $\phi_n$ decrease with $n$ in an order of $1/n$, which is very slow and a large number of terms are required to approximate it well enough. This also explains that spectral method is not suitable to deal with strong segregation systems where the density profile resembles a square wave function.