# Eigenfunction

In mathematics, an **eigenfunction** of a linear operator, Template:Mvar, 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

for some scalar, Template:Mvar, 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 *λ* = *λ _{n}* (

*n*= 1, 2, 3, ...) that admit a corresponding solution for

*f*=

*f*(with each

_{n}*f*belonging to the eigenvalue

_{n}*λ*) when combined with the boundary conditions. Eigenfunctions are used to analyze Template:Mvar.

_{n}For example, *f _{k}* (

*x*) =

*e*is an eigenfunction for the differential operator

^{kx}for any value of Template:Mvar, with corresponding eigenvalue *λ* = *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 .

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*) = *λ f* (*t*) with the complex constant Template:Mvar.^{[1]}

## Examples

### Derivative operator

A widely used class of linear operators acting on function spaces are the differential operators on function spaces. As an example, on the space *C*^{∞} of infinitely differentiable real functions of a real argument Template:Mvar, the process of differentiation is a linear operator since

The eigenvalue equation for a linear differential operator Template:Mvar in *C*^{∞} is then a differential equation

The functions that satisfy this equation are commonly called **eigenfunctions**. For the derivative operator {{ safesubst:#invoke:Unsubst||$B=*d*/*dt*}}, an eigenfunction is a function that, when differentiated, yields a constant times the original function. That is,

for all Template:Mvar. This equation can be solved for any value of Template:Mvar. The solution is an exponential function

The derivative operator is defined also for complex-valued functions of a complex argument. In the complex version of the space *C*^{∞}, the eigenvalue equation has a solution for any complex constant Template:Mvar. The spectrum of the operator {{ safesubst:#invoke:Unsubst||$B=*d*/*dt*}} is therefore the whole complex plane. This is an example of a continuous spectrum.

## Applications

### Vibrating strings

Let *h*(*x*, *t*) denote the sideways displacement of a stressed elastic chord, such as the vibrating strings of a string instrument, as a function of the position Template:Mvar along the string and of time Template:Mvar. From the laws of mechanics, applied to infinitesimal portions of the string, one can deduce that the function Template:Mvar satisfies the partial differential equation

which is called the (one-dimensional) wave equation. Here Template:Mvar is a constant that depends on the tension and mass of the string.

This problem is amenable to the method of separation of variables. If we assume that *h*(*x*, *t*) can be written as the product of the form *X*(*x*)*T*(*t*), we can form a pair of ordinary differential equations:

Each of these is an eigenvalue equation, for eigenvalues and −*ω*^{2}, respectively. For any values of Template:Mvar and Template:Mvar, the equations are satisfied by the functions

where Template:Mvar and Template:Mvar are arbitrary real constants. If we impose boundary conditions (that the ends of the string are fixed with *X*(*x*) = 0 at *x* = 0 and *x* = *L*, for example) we can constrain the eigenvalues. For those boundary conditions, we find sin(*φ*) = 0, and so the phase angle *φ* = 0 and

Thus, the constant Template:Mvar is constrained to take one of the values *ω _{n}* = {{ safesubst:#invoke:Unsubst||$B=

*ncπ*/

*L*}}, where Template:Mvar is any integer. Thus, the clamped string supports a family of standing waves of the form

From the point of view of our musical instrument, the frequency *ω _{n}* is the frequency of the Template:Mvar-th harmonic, which is called the (

*n*− 1)-th overtone.

### Quantum mechanics

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

with

has solutions of the form

where *φ _{k}* are eigenfunctions of the operator with eigenvalues

*E*. The fact that only certain eigenvalues

_{k}*E*with associated eigenfunctions

_{k}*φ*satisfy Schrödinger's equation leads to a natural basis for quantum mechanics and the periodic table of the elements, with each

_{k}*E*an allowable energy state of the system. The success of this equation in explaining the spectral characteristics of hydrogen is considered one of the greatest triumphs of 20th century physics.

_{k}Since the Hamiltonian operator is a Hermitian Operator, its eigenfunctions are orthogonal functions. This is not necessarily the case for eigenfunctions of other operators (such as the example Template:Mvar mentioned above). Orthogonal functions *f _{i}* (

*i*= 1, 2, ...) have the property that

where Template:Overline is the complex conjugate of *f _{i}* .

whenever *i* ≠ *j*, in which case the set { *f _{i}* |

*i*∈

*I*} is said to be orthogonal. Also, it is linearly independent.

## Notes

- ↑ Bernd Girod, Rudolf Rabenstein, Alexander Stenger,
*Signals and systems*, 2nd ed., Wiley, 2001, ISBN 0-471-98800-6 p. 49

## 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)