eigenfunctions

An operator takes functions and outputs a function. For example, the laplacian is an operator that takes a function and outputs a function that gives the summation of all second derivatives involving one variable.

If you have an eigenfunction \(f\) of an operator, then hitting \(f\) with the operator returns a multiple of \(f\).

For example, \(\sin(\omega x)\) is an eigenfunction of the Laplacian, because the \(\frac{d\sin(\omega x)}{dx} = -\omega^2 \sin(\omega x)\).