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

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

For example, f_k(x) = e^kx is an eigenfunction for the differential operator

for any value of k, with a corresponding eigenvalue λ = k² - k.

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

has solutions of the form

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