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The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices?

In particular:

Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual combinatorial laplacian is:

$$ \Delta f(x,y) = 4 f(x,y) - f(x+1,y) - f(x-1,y) - f(x,y+1) - f(x,y-1) \; . $$

Suppose we also add the diagonals:

$$ \Delta f(x,y) = 6 f(x,y) - f(x+1,y) - f(x-1,y) - f(x,y+1) - f(x,y-1) - f(x+1,y+1) - f(x-1,y-1) \; . $$

(In some sense, this corresponds to a smooth laplacian with off diagonal metric element.) Does anybody know the spectrum for this?

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2  
Do Fourier transform – Alexander Chervov Jan 16 '13 at 19:45
1  
Alexander is perfectly correct, let me just give a hint: the spectrum is the set of values of the function $6-2\cos(x)-2\cos(y)-2\cos(x+y)$ on the 2-torus. I'll leave the rest to you! – Alain Valette Jan 16 '13 at 21:00

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