eigenfunction

An eigenfunction of a linear operator T on a vector space of functions is a nonzero function f for which T f = λ f for some scalar λ, called an eigenvalue. This generalizes the notion of an eigenvector for matrices to function spaces.

Where T: V -> V is linear, the eigenfunctions are the functions whose shape is preserved by T up to a scale. The set of all eigenfunctions corresponding to a particular eigenvalue λ forms the eigenspace.

Common examples: For the differentiation operator D f = f', on suitable function spaces, the exponential functions f(x) = e^{r x} satisfy D f = r f, so they are eigenfunctions with eigenvalue r. If a linear differential operator L has constant coefficients, exponential functions e^{r x} often appear as eigenfunctions. In a spatial domain, the second-derivative operator d^2/dx^2 has eigenfunctions sin(k x) and cos(k x) under periodic or defined boundary conditions, with eigenvalues -k^2; more generally, Sturm–Liouville problems yield eigenfunctions that satisfy L y = λ w y with orthogonality properties relative to a weight function w.

On finite-dimensional spaces, T corresponds to a matrix and eigenfunctions reduce to eigenvectors. In infinite-dimensional spaces, the spectrum may be discrete (eigenvalues with normalizable eigenfunctions) or continuous.

Properties: if T is self-adjoint (or unitary), eigenfunctions corresponding to distinct eigenvalues are orthogonal. Eigenfunctions form a basis under appropriate conditions, enabling expansions of functions in eigenfunction series such as Fourier or Sturm–Liouville expansions.

Applications appear in quantum mechanics, vibration analysis, and solving linear PDEs. An eigenfunction is a function that remains essentially the same under the action of an operator, up to a scalar multiple.