egenfunktion

An eigenfunction is a nonzero function that, when a linear operator acts on it, is scaled by a constant called the eigenvalue. More precisely, for a linear operator T acting on a space of functions, a function f is an eigenfunction with eigenvalue λ if T f = λ f. The pair (f, λ) is called an eigenpair. Eigenfunctions generalize the notion of eigenvectors from finite-dimensional vector spaces to function spaces.

Examples and contexts help illustrate the concept. For the differentiation operator D, defined by (D f)(x) = f′(x) on a suitable space of differentiable functions, the exponential functions f(x) = e^{λ x} are eigenfunctions with eigenvalue λ, since D e^{λ x} = λ e^{λ x}. For integral operators, such as (T f)(x) = ∫ K(x, y) f(y) dy, eigenfunctions satisfy the integral equation ∫ K(x, y) φ(y) dy = λ φ(x). In finite dimensions, if a linear transformation is represented by a matrix A, the eigenvectors of A correspond to eigenfunctions in the discrete setting.

In applied contexts, eigenfunctions arise in Sturm–Liouville problems, where solutions y(x) satisfy (L y)(x) = λ w(x) y(x) with boundary conditions, producing a spectrum of eigenvalues and orthogonal eigenfunctions with respect to a weight function w. In Hilbert spaces, self-adjoint operators have real eigenvalues and their eigenfunctions associated with distinct eigenvalues are orthogonal; under suitable conditions they form a complete basis, enabling expansions like Fourier series in appropriate problems.