Inmathematics, aneigenfunction of alinear operatorD defined on somefunction space is any non-zerofunction${\displaystyle f}$ in that space that, when acted upon byD, is only multiplied by some scaling factor called aneigenvalue. As an equation, this condition can be written as

$${\displaystyle Df=\lambda f}$$for somescalar eigenvalue${\displaystyle \lambda .}$^[1][2][3] The solutions to this equation may also be subject toboundary conditions that limit the allowable eigenvalues and eigenfunctions.

An eigenfunction is a type ofeigenvector.

In general, an eigenvector of a linear operatorD defined on somevector space is a nonzero vector in the domain ofD that, whenD acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case whereD is defined on a function space, the eigenvectors are referred to aseigenfunctions. That is, a functionf is an eigenfunction ofD if it satisfies the equation

$${\displaystyle Df=\lambda f,}$$

1

where λ is a scalar.^[1][2][3] The solutions to Equation (1) may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete setλ₁,λ₂, … or to a continuous set over some range. The set of all possible eigenvalues ofD is sometimes called itsspectrum, which may be discrete, continuous, or a combination of both.^[1]

Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to bedegenerate and the maximum number of linearly independent eigenfunctions associated with the same eigenvalue is the eigenvalue'sdegree of degeneracy orgeometric multiplicity.^[4][5]

A widely used class of linear operators acting on infinite dimensional spaces are differential operators on the spaceC^∞ of infinitely differentiable real or complex functions of a real or complex argumentt. For example, consider the derivative operator$\frac{d}{dt}$ with eigenvalue equation

$${\displaystyle \frac{d}{dt}f(t)=\lambda f(t).}$$

This differential equation can be solved by multiplying both sides by$\frac{dt}{f(t)}$ and integrating. Its solution, theexponential function

$${\displaystyle f(t)=f_0{e}^{\lambda t},}$$

is the eigenfunction of the derivative operator, wheref₀ is a parameter that depends on the boundary conditions. Note that in this case the eigenfunction is itself a function of its associated eigenvalue λ, which can take any real or complex value. In particular, note that for λ = 0 the eigenfunctionf(t) is a constant.

Suppose in the example thatf(t) is subject to the boundary conditionsf(0) = 1 and${\frac{df}{dt}|}_{t=0}=2$. We then find that

$${\displaystyle f(t)=e^{2t},}$$

where λ = 2 is the only eigenvalue of the differential equation that also satisfies the boundary condition.

Eigenfunctions can be expressed as column vectors and linear operators can be expressed as matrices, although they may have infinite dimensions. As a result, many of the concepts related to eigenvectors of matrices carry over to the study of eigenfunctions.

Define theinner product in the function space on whichD is defined as

$${\displaystyle ⟨f,g⟩={\int }_{\mathrm{\Omega }}{f}^{\ast }(t)g(t)dt,}$$

integrated over some range of interest fort called Ω. The* denotes thecomplex conjugate.

Suppose the function space has anorthonormal basis given by the set of functions {u₁(t),u₂(t), …,u_n(t)}, wheren may be infinite. For the orthonormal basis,

whereδ_ij is theKronecker delta and can be thought of as the elements of theidentity matrix.

Functions can be written as alinear combination of the basis functions,

$${\displaystyle f(t)=\sum _{j=1}^n{b}_j{u}_j(t),}$$

for example through aFourier expansion off(t). The coefficientsb_j can be stacked into ann by 1 column vectorb = [b₁b₂ …b_n]^T. In some special cases, such as the coefficients of the Fourier series of a sinusoidal function, this column vector has finite dimension.

Additionally, define a matrix representation of the linear operatorD with elements

$${\displaystyle A_{ij}=⟨u_i,D{u}_j⟩={\int }_{\mathrm{\Omega }}{u}_i^{\ast }(t)D{u}_j(t)dt.}$$

We can write the functionDf(t) either as a linear combination of the basis functions or asD acting upon the expansion off(t),

$${\displaystyle Df(t)=\sum _{j=1}^n{c}_j{u}_j(t)=\sum _{j=1}^n{b}_{j}D{u}_j(t).}$$

Taking the inner product of each side of this equation with an arbitrary basis functionu_i(t),

This is the matrix multiplicationAb =c written in summation notation and is a matrix equivalent of the operatorD acting upon the functionf(t) expressed in the orthonormal basis. Iff(t) is an eigenfunction ofD with eigenvalue λ, thenAb =λb.

Many of the operators encountered in physics areHermitian. Suppose the linear operatorD acts on a function space that is aHilbert space with an orthonormal basis given by the set of functions {u₁(t),u₂(t), …,u_n(t)}, wheren may be infinite. In this basis, the operatorD has a matrix representationA with elements

$${\displaystyle A_{ij}=⟨u_i,D{u}_j⟩={\int }_{\mathrm{\Omega }}dt{u}_i^{\ast }(t)D{u}_j(t).}$$

integrated over some range of interest fort denoted Ω.

By analogy withHermitian matrices,D is a Hermitian operator ifA_ij =A_ji*, or:^[6]

Consider the Hermitian operatorD with eigenvaluesλ₁,λ₂, ... and corresponding eigenfunctionsf₁(t),f₂(t), …. This Hermitian operator has the following properties:

• Its eigenvalues are real,λ_i =λ_i*^[4][6]
• Its eigenfunctions obey an orthogonality condition,${\displaystyle ⟨f_i,f_j⟩=0}$ ifi ≠j^[6][7][8]

The second condition always holds forλ_i ≠λ_j. For degenerate eigenfunctions with the same eigenvalueλ_i, orthogonal eigenfunctions can always be chosen that span the eigenspace associated withλ_i, for example by using theGram-Schmidt process.^[5] Depending on whether the spectrum is discrete or continuous, the eigenfunctions can be normalized by setting the inner product of the eigenfunctions equal to either a Kronecker delta or aDirac delta function, respectively.^[8][9]

For many Hermitian operators, notablySturm–Liouville operators, a third property is

• Its eigenfunctions form a basis of the function space on which the operator is defined^[5]

As a consequence, in many important cases, the eigenfunctions of the Hermitian operator form an orthonormal basis. In these cases, an arbitrary function can be expressed as a linear combination of the eigenfunctions of the Hermitian operator.

Leth(x,t) denote the transverse displacement of a stressed elastic chord, such as thevibrating strings of astring instrument, as a function of the positionx along the string and of timet. Applying the laws of mechanics toinfinitesimal portions of the string, the functionh satisfies thepartial differential equation

$${\displaystyle \frac{{\mathrm{\partial }}^{2}h}{\mathrm{\partial }{t}^2}=c^2\frac{{\mathrm{\partial }}^{2}h}{\mathrm{\partial }{x}^2},}$$

which is called the (one-dimensional)wave equation. Herec is a constant speed that depends on the tension and mass of the string.

This problem is amenable to the method ofseparation of variables. If we assume thath(x,t) can be written as the product of the formX(x)T(t), we can form a pair of ordinary differential equations:

$${\displaystyle \frac{d^2}{d{x}^2}X=-\frac{{\omega }^2}{c^2}X,\frac{d^2}{d{t}^2}T=-{\omega }^{2}T.}$$

Each of these is an eigenvalue equation with eigenvalues

$-\frac{{\omega }^2}{c^2}$ and−ω², respectively. For any values ofω andc, the equations are satisfied by the functions

$${\displaystyle X(x)=\sin \left(\frac{\omega x}{c}+\phi \right),T(t)=\sin (\omega t+\psi ),}$$where the phase anglesφ andψ are arbitrary real constants.

If we impose boundary conditions, for example that the ends of the string are fixed atx = 0 andx =L, namelyX(0) =X(L) = 0, and thatT(0) = 0, we constrain the eigenvalues. For these boundary conditions,sin(φ) = 0 andsin(ψ) = 0, so the phase anglesφ =ψ = 0, and

$${\displaystyle \sin \left(\frac{\omega L}{c}\right)=0.}$$

This last boundary condition constrainsω to take a valueω_n =⁠ncπ/L⁠, wheren is anyinteger. Thus, the clamped string supports a family of standing waves of the form

$${\displaystyle h(x,t)=\sin \left(\frac{n\pi x}{L}\right)\sin ({\omega }_{n}t).}$$

In the example of a string instrument, the frequencyω_n is the frequency of then-thharmonic, which is called the(n − 1)-thovertone.

Inquantum mechanics, theSchrödinger equation

$${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }(\mathbf{r},t)=H\mathrm{\Psi }(\mathbf{r},t)}$$

with theHamiltonian operator

$${\displaystyle H=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V(\mathbf{r},t)}$$can be solved by separation of variables if the Hamiltonian does not depend explicitly on time.^[10] In that case, thewave functionΨ(r,t) =φ(r)T(t) leads to the two differential equations,

$${\displaystyle H\phi (\mathbf{r})=E\phi (\mathbf{r}),}$$

2

$${\displaystyle i\hslash \frac{\mathrm{\partial }T(t)}{\mathrm{\partial }t}=ET(t).}$$

3

Both of these differential equations are eigenvalue equations with eigenvalueE. As shown in an earlier example, the solution of Equation (3) is the exponential$${\displaystyle T(t)=e^{-iEt/\hslash }.}$$

Equation (2) is the time-independent Schrödinger equation. The eigenfunctionsφ_k of the Hamiltonian operator arestationary states of the quantum mechanical system, each with a corresponding energyE_k. They represent allowable energy states of the system and may be constrained by boundary conditions.

The Hamiltonian operatorH is an example of a Hermitian operator whose eigenfunctions form an orthonormal basis. When the Hamiltonian does not depend explicitly on time, general solutions of the Schrödinger equation are linear combinations of the stationary states multiplied by the oscillatoryT(t),^[11]$\mathrm{\Psi }(\mathbf{r},t)=\sum _k{c}_k{\phi }_k(\mathbf{r})e^{-i{E}_{k}t/\hslash }$ or, for a system with a continuous spectrum,

$${\displaystyle \mathrm{\Psi }(\mathbf{r},t)=\int dE{c}_E{\phi }_E(\mathbf{r})e^{-iEt/\hslash }.}$$

The success of the Schrödinger equation in explaining the spectral characteristics of hydrogen is considered one of the greatest triumphs of 20th century physics.

In the study ofsignals and systems, an eigenfunction of a system is a signalf(t) that, when input into the system, produces a responsey(t) =λf(t), whereλ is a complex scalar eigenvalue.^[12]

• Eigenvalues and eigenvectors
• Fixed point combinator
• Fourier transform eigenfunctions
• Hilbert–Schmidt theorem
• Spectral theory of ordinary differential equations

• Functional analysis

• Articles with short description
• Short description is different from Wikidata