Eigenfunction

This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk.

In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has

$\mathcal A f = \lambda f$

for some scalar, λ, the corresponding eigenvalue. The solution of the differential eigenvalue problem also depends on any boundary conditions required of $f$. In each case there are only certain eigenvalues $\lambda=\lambda_n$ ($n=1,2,3,...$) that admit a corresponding solution for $f=f_n$ (with each $f_n$ belonging to the eigenvalue $\lambda_n$) when combined with the boundary conditions. The existence of eigenfunctions is typically the most insightful way to analyze $A$.

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

$\mathcal A = \frac{d^2}{dx^2} - \frac{d}{dx}$

for any value of $k$, with corresponding eigenvalue $\lambda = k^2 - k$. If boundary conditions are applied to this system (e.g., $f=0$ at two physical locations in space), then only certain values of $k=k_n$ satisfy the boundary conditions, generating corresponding discrete eigenvalues $\lambda_n=k_n^2-k_n$.

Specifically, in the study of signals and systems, the eigenfunction of a system is the signal $f(t)$ which when input into the system, produces a response $y(t) = \lambda f(t)$ with the complex constant $\lambda$.

Applications

Eigenfunctions play an important role in many branches of physics. An important example is quantum mechanics, where the Schrödinger equation

$\mathcal H \psi = E \psi$,

with

$\mathcal H = -\frac{\hbar^2}{2m}\nabla^2+ V(\mathbf{r},t)$

has solutions of the form

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

where $\phi_k$ are eigenfunctions of the operator $\mathcal H$ with eigenvalues $E_k$. The fact that only certain eigenvalues $E_k$ with associated eigenfunctions $\phi_k$ satisfy Schrödinger's equation leads to a natural basis for quantum mechanics and the periodic table of the elements, with each $E_k$ an allowable energy state of the system. The success of this equation in explaining the spectral characteristics of hydrogen is considered one of the great triumphs of 20th century physics.

Due to the nature of the Hamiltonian operator $\mathcal H$, its eigenfunctions are orthogonal functions. This is not necessarily the case for eigenfunctions of other operators (such as the example $A$ mentioned above). Orthogonal functions $f_i$, $i=1, 2, \dots,$ have the property that

$0 = \int f_i^{*} f_j$

where $f_i^{*}$ is the complex conjugate of $f_i$

whenever $i\neq j$, in which case the set $\{f_i \,|\, i \in I\}$ is said to be orthogonal. Also, it is linearly independent.

References

• Methods of Mathematical Physics by R. Courant, D. Hilbert ISBN 0-471-50447-5 (Volume 1 Paperback) ISBN 0-471-50439-4 (Volume 2 Paperback) ISBN 0-471-17990-6 (Hardback)

See also

• Eigenvalue, eigenvector and eigenspace
• Hilbert–Schmidt theorem
• Spectral theory of ordinary differential equations
• Fixed point combinator