Eigenfunction

An eigenfunction is a nonzero function $f$ satisfying $Lf=\lambda f$ for a linear operator $L$ and scalar eigenvalue $\lambda$.

A nodeless real eigenfunction has no zero in the interior of the domain. For a one-dimensional confining Schrödinger operator, the nodeless theorem for a one-dimensional ground state identifies the ground state as the eigenfunction that can be chosen strictly positive. Normalizable excited states have nodes.

A Dirichlet eigenfunction is an eigenfunction of a differential operator satisfying a homogeneous Dirichlet boundary condition. The term applies to more general boundary-value operators as well as the Dirichlet Laplacian.

An adjoint eigenfunction is a nonzero eigenfunction of the adjoint operator on its adjoint domain. For an inner product conjugate-linear in its first argument, $L^{\ast }g=\overline{\lambda }g$ implies $⟨g,(L-\lambda )f⟩=0$ for every $f$ in the domain of $L$. Consequently a necessary solvability condition for $(L-\lambda )f=h$ is $⟨g,h⟩=0$. If $L-\lambda I$ is a Fredholm operator, orthogonality to every element of its adjoint kernel is also sufficient. Boundary conditions and weights in a coupled system enter the adjoint domain and must be derived by integration by parts; copying the original eigenfunction without checking them can give a wrong projection.

A generalized eigenfunction solves the eigenvalue equation for a linear operator in an enlarged space, often a space of distributions, although it may fail to belong to the underlying Hilbert space. A plane wave $e^{ikx}$ on the whole real line is a generalized momentum eigenstate, with momentum $\hslash k$, and is not square-integrable. This usage differs from a generalized eigenvector in a Jordan chain, where a higher power of $L-\lambda$ vanishes rather than $L-\lambda$ itself.