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Lattice Waves -- Translational Invariance

In the previous example , a traditional separation of variables was possible because the translations occurred in the direction of the coordinate axes. Here, we construct eigenfunctions of the Laplacian that are periodic with respect to any lattice.

This picture shows a wallpaper function with translations in two directions that are not the simple coordinate directions. The idea here is to develop coordinates adapted to the lattice.

Suppose we wish to construct eigenfunctions periodic with respect to translations along two linearly independent vectors t₁ and t₂, and therefore along every vector in the lattice generated by these two.

Consider complex exponential functions of the form f(x) = e^(2pix·w), familiar from physics as standing waves with level sets perpendicular to the vector w. Direct computation shows that every such wave is an eigenfunction of the Laplacian with eigenvalue -4 p²(w·w). What choices for w will produce the desired periodicity?

Since we require f(x+t_i) = f(x) , one quickly sees that a necessary and sufficient condition is that both values t_i·w should be integers. The set of vectors w satisfying this condition form a lattice of their own, called the dual lattice of the original one. (We learned this in an appendix to Symmetry and Chaos [Fi], an interesting book showing how to construct chaotic systems with various symmetries.) This lattice is generated by vectors k₁ and k₂ defined by the equations t_i·k_j = d_ij, where d_ij is zero or one as i and j are different or the same.

The most general eigenfunction periodic with respect to the lattice generated by t₁ and t₂ is

where n and m are integers, not necessarily positive.

For computational purposes, it will be convenient to expand this formula in terms of the components of k₁ and k₂. Say k₁ = (a,b) and k₂ = (c,d). Then

where X = 2p(xa + yb) and Y = 2p(xc + yd). We call X and Y the lattice coordinates for the lattice. They amount to new variables measuring displacements along the lattice vectors.

Since, at this point, we are interested in real-valued functions, we now have cos(nX+mY) and sin(nX+mY), the real and imaginary parts of the complex wave above, as the only eigenfunctions periodic with respect to the given lattice. We call these functions lattice waves.

The basic result of Fourier analysis (see [Ga]), is that any function (say a continuous one) periodic with respect to the given lattice can be constructed as a superposition of these lattice waves. Thus, every recipe for wallpaper functions starts with these ingredients. To see what is involved in imposing other symmetries, continue with the example in the hexagonal lattice .

Eigenfunctions with 3-Fold Rotations Experiment interactively with Wallpaper Design Constructing Wallpaper Functions Example in a Rectangular Lattice