In mathematics, an eigenfunction f of a linear operator A on a function space is an eigenvector of the linear operator; it satisfies

<math>

\mathcal A f = \lambda f <math>

for some scalar λ, the corresponding eigenvalue. The existence of eigenvectors is typically a great help in analysing A.

For example, <math>f_k(x) = e^{kx}<math> is an eigenfunction for the differential operator

<math> \mathcal A = \frac{d^2}{dx^2} - \frac{d}{dx}, <math>

for any value of <math>k<math>, with a corresponding eigenvalue <math>\lambda = k^2 - k<math>.

Eigenfunctions play an important role in quantum mechanics, where the Schrödinger equation

<math>

i \hbar \frac{\partial}{\partial t} \psi = \mathcal H \psi <math>

has solutions of the form

<math>

\psi(t) = \sum_k e^{-i E_k t/\hbar} \phi_k, <math>

where <math>\phi_k<math> are eigenfunctions of the operator <math>\mathcal H<math> with eigenvalues <math>E_k<math>. Due to the nature of the hamiltonian operator <math>\mathcal H<math>, its eigenfunctions are orthogonal functions. This is not necessarily the case for eigenfunctions of other operators (such as the example <math>A<math> mentioned above)