n this section we compute the spectrum of the Laplacian
explicitly. It will be seen that the Gershgorin theorem gives a good boundary.
We introduce the lattice
and consider the spectral
problem


(Laplace problem)



(Laplace bounds)

We introduce the lattice functions
with the
values
Such functions satisfy the boundary conditions
(
Laplace bounds
) and constitute a basis in
linear algebra sense. We compute action of the operator
on
:
Since
we have
Therefore,
are eigenvalues and
constitute the spectrum of
:


(Laplacian Spectrum)

We have the spectral
limits


(Laplacian Limits)

