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Applying the Laplacian operator to a function usually results in a new function totally different from the original. For example, the Laplacian of x2 + y2 is 4.
For very special functions, f, however, the Laplacian of f is exactly a multiple of f. These are called eigenfunctions of the Laplacian, and the multiple is called an eigenvalue, just as in finite-dimensional linear algebra. The set of all eigenvalues for a function is called the spectrum of the function.
For a low-dimensional example, think of a single-variable function whose second derivative is proportional to the original function. Two familiar examples are the sine and cosine functions, but the hyperbolic trigonometric functions work too, as do simple exponential functions. In many problems, we want a function that is periodic with a particular period. In such a case, the hyperbolic and exponential functions would be rejected and the set of eigenvalues is discrete. When the period is set to be 2p, the spectrum is simply the set of negatives of squares of integers .
This is quite similar to what happens when we look for eigenfunctions of the Laplacian that are periodic with respect to a lattice. Test for yourself that the function sin(x)cos(y) is an eigenfunction of the planar Laplacian with eigenvalue -1, periodic with respect to translations by 2p both up and down and side to side.
In every case we consider, we end up with a discrete spectrum and a finite-dimensional space of eigenfunctions for each eigenvalue. An example in a rectangular lattice illustrates this .