Editing Eigenfunction

{{short description|Mathematical function of a linear operator}}
[[File:Drum vibration mode12.gif|right|frame|This solution of the [[vibrations of a circular drum|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 map|linear operator]] ''D'' defined on some [[function space]] is any non-zero [[function (mathematics)|function]] <math>f</math> in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an [[eigenvalues and eigenvectors|eigenvalue]]. As an equation, this condition can be written as
<math display="block">Df = \lambda f</math>
for some [[scalar (mathematics)|scalar]] eigenvalue <math>\lambda.</math>{{sfn|Davydov|1976|p=20}}{{sfn|Kusse|Westwig|1998|p=435}}{{sfn|Wasserman|2016}} The solutions to this equation may also be subject to [[Boundary value problem#boundary value conditions|boundary conditions]] that limit the allowable eigenvalues and eigenfunctions.

An eigenfunction is a type of [[eigenvalues and eigenvectors|eigenvector]].

==Eigenfunctions==
In general, an eigenvector of a linear operator ''D'' defined on some vector space is a nonzero vector in the domain of ''D'' that, when ''D'' acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where ''D'' is defined on a function space, the eigenvectors are referred to as '''eigenfunctions'''. That is, a function ''f'' is an eigenfunction of ''D'' if it satisfies the equation
{{NumBlk||<math display="block">Df = \lambda f,</math>|{{EquationRef|1}}}}
where λ is a scalar.{{sfn|Davydov|1976|p=20}}{{sfn|Kusse|Westwig|1998|p=435}}{{sfn|Wasserman|2016}} The solutions to Equation {{EqNote|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 set ''λ''<sub>1</sub>, ''λ''<sub>2</sub>, … or to a continuous set over some range. The set of all possible eigenvalues of ''D'' is sometimes called its [[Spectrum (functional analysis)|spectrum]], which may be discrete, continuous, or a combination of both.{{sfn|Davydov|1976|p=20}}

Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be [[Degenerate energy levels#mathematics|degenerate]] and the maximum number of linearly independent eigenfunctions associated with the same eigenvalue is the eigenvalue's ''degree of degeneracy'' or [[Eigenvalues and eigenvectors#Eigenspaces, geometric multiplicity, and the eigenbasis|geometric multiplicity]].{{sfn|Davydov|1976|p=21}}{{sfn|Kusse|Westwig|1998|p=437}}

===Derivative example===
A widely used class of linear operators acting on infinite dimensional spaces are differential operators on the space '''C'''<sup>∞</sup> of infinitely differentiable real or complex functions of a real or complex argument ''t''. For example, consider the derivative operator <math display="inline" alt="d over dt">\frac{d}{dt}</math> with eigenvalue equation
<math display="block" alt="the derivative of f of t equals lambda times f of t"> \frac{d}{dt}f(t) = \lambda f(t).</math>

This differential equation can be solved by multiplying both sides by <math display="inline" alt="dt over f of t">\frac{dt}{f(t)}</math> and integrating. Its solution, the [[exponential function]]
<math display="block" alt="f of t equals f nought times e raised to lambda t"> f(t)=f_0 e^{\lambda t},</math>
is the eigenfunction of the derivative operator, where ''f''<sub>0</sub> 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 eigenfunction ''f''(''t'') is a constant.

Suppose in the example that ''f''(''t'') is subject to the boundary conditions ''f''(0) = 1 and <math display="inline" alt="df over dt at t equals 0 is 2">\left.\frac{df}{dt}\right|_{t=0} = 2</math>. We then find that
<math display="block" alt="f of t equals e raised to 2t"> f(t)=e^{2t},</math>
where λ = 2 is the only eigenvalue of the differential equation that also satisfies the boundary condition.

===Link to eigenvalues and eigenvectors of matrices===
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 the [[inner product]] in the function space on which ''D'' is defined as
<math display="block"> \langle f,g \rangle = \int_{\Omega} \ f^*(t)g(t) dt,</math>
integrated over some range of interest for ''t'' called Ω. The ''*'' denotes the [[complex conjugate]].

Suppose the function space has an [[orthonormal basis]] given by the set of functions {''u''<sub>1</sub>(''t''), ''u''<sub>2</sub>(''t''), …, ''u''<sub>''n''</sub>(''t'')}, where ''n'' may be infinite. For the orthonormal basis,
<math display="block"> \langle u_i,u_j \rangle = \int_{\Omega} \ u_i^*(t)u_j(t) dt = \delta_{ij} =
\begin{cases}
  1 & i=j \\
  0 & i \ne j
\end{cases},</math>
where ''δ''<sub>''ij''</sub> is the [[Kronecker delta]] and can be thought of as the elements of the [[identity matrix]].

Functions can be written as a linear combination of the basis functions,
<math display="block">f(t) = \sum_{j=1}^n b_j u_j(t),</math>
for example through a [[Fourier series|Fourier expansion]] of ''f''(''t''). The coefficients ''b''<sub>''j''</sub> can be stacked into an ''n'' by 1 column vector {{nowrap|1=''b'' = [''b''<sub>1</sub> ''b''<sub>2</sub> … ''b''<sub>''n''</sub>]<sup>T</sup>}}. 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 operator ''D'' with elements
<math display="block"> A_{ij} = \langle u_i,Du_j \rangle = \int_{\Omega}\ u^*_i(t)Du_j(t) dt.</math>

We can write the function ''Df''(''t'') either as a linear combination of the basis functions or as ''D'' acting upon the expansion of ''f''(''t''),
<math display="block">Df(t) = \sum_{j=1}^n c_j u_j(t) = \sum_{j=1}^n b_j Du_j(t).</math>

Taking the inner product of each side of this equation with an arbitrary basis function ''u''<sub>''i''</sub>(''t''),
<math display="block">\begin{align}
  \sum_{j=1}^n c_j \int_{\Omega} \ u_i^*(t)u_j(t) dt &= \sum_{j=1}^n b_j \int_{\Omega} \ u_i^*(t)Du_j(t) dt, \\
  c_i &= \sum_{j=1}^n b_j A_{ij}.
\end{align}</math>

This is the matrix multiplication ''Ab'' = ''c'' written in summation notation and is a matrix equivalent of the operator ''D'' acting upon the function ''f''(''t'') expressed in the orthonormal basis. If ''f''(''t'') is an eigenfunction of ''D'' with eigenvalue λ, then ''Ab'' = ''λb''.

===Eigenvalues and eigenfunctions of Hermitian operators===
Many of the operators encountered in physics are [[self-adjoint operator|Hermitian]]. Suppose the linear operator ''D'' acts on a function space that is a [[Hilbert space]] with an orthonormal basis given by the set of functions {''u''<sub>1</sub>(''t''), ''u''<sub>2</sub>(''t''), …, ''u''<sub>''n''</sub>(''t'')}, where ''n'' may be infinite. In this basis, the operator ''D'' has a matrix representation ''A'' with elements
<math display="block"> A_{ij} = \langle u_i,Du_j \rangle = \int_{\Omega} dt\ u^*_i(t)Du_j(t).</math>
integrated over some range of interest for ''t'' denoted Ω.

By analogy with [[Hermitian matrix|Hermitian matrices]], ''D'' is a Hermitian operator if ''A''<sub>''ij''</sub> = ''A''<sub>''ji''</sub>*, or:{{sfn|Kusse|Westwig|1998|p=436}} <math display="block">\begin{align}
\langle u_i,Du_j \rangle &= \langle Du_i,u_j \rangle, \\[-1pt]
\int_{\Omega} dt\ u^*_i(t)Du_j(t) &= \int_{\Omega} dt\ u_j(t)[Du_i(t)]^*.
\end{align}</math>

Consider the Hermitian operator ''D'' with eigenvalues ''λ''<sub>1</sub>, ''λ''<sub>2</sub>, … and corresponding eigenfunctions ''f''<sub>1</sub>(''t''), ''f''<sub>2</sub>(''t''), …. This Hermitian operator has the following properties:
* Its eigenvalues are real, ''λ''<sub>''i''</sub> = ''λ''<sub>''i''</sub>*{{sfn|Davydov|1976|p=21}}{{sfn|Kusse|Westwig|1998|p=436}}
* Its eigenfunctions obey an orthogonality condition, <math alt="inner product of f sub i and f sub j equals 0">\langle f_i,f_j \rangle = 0 </math> if ''i'' ≠ ''j''{{sfn|Kusse|Westwig|1998|p=436}}{{sfn|Davydov|1976|p=24}}{{sfn|Davydov|1976|p=29}}

The second condition always holds for ''λ''<sub>''i''</sub> ≠ ''λ''<sub>''j''</sub>. For degenerate eigenfunctions with the same eigenvalue ''λ''<sub>''i''</sub>, orthogonal eigenfunctions can always be chosen that span the eigenspace associated with ''λ''<sub>''i''</sub>, for example by using the [[Gram-Schmidt process]].{{sfn|Kusse|Westwig|1998|p=437}} 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 a [[Dirac delta function]], respectively.{{sfn|Davydov|1976|p=29}}{{sfn|Davydov|1976|p=25}}

For many Hermitian operators, notably [[Sturm–Liouville theory|Sturm–Liouville operators]], a third property is
* Its eigenfunctions form a basis of the function space on which the operator is defined{{sfn|Kusse|Westwig|1998|p=437}}

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.

==Applications==
===Vibrating strings===
[[File:Standing wave.gif|thumb|270px|The shape of a standing wave in a string fixed at its boundaries is an example of an eigenfunction of a differential operator. The admissible eigenvalues are governed by the length of the string and determine the frequency of oscillation.]]
Let {{math|''h''(''x'', ''t'')}} denote the transverse displacement of a stressed elastic chord, such as the [[vibrating string]]s of a [[string instrument]], as a function of the position {{mvar|x}} along the string and of time {{mvar|t}}. Applying the laws of mechanics to [[infinitesimal]] portions of the string, the function {{mvar|h}} satisfies the [[partial differential equation]]
<math display="block" alt="the second partial derivative of h with respect to t equals c squared times the second partial derivative of h with respect to x">\frac{\partial^2 h}{\partial t^2} = c^2\frac{\partial^2 h}{\partial x^2},</math>
which is called the (one-dimensional) [[wave equation]]. Here {{mvar|c}} is a constant speed 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 {{math|''h''(''x'', ''t'')}} can be written as the product of the form {{math|''X''(''x'')''T''(''t'')}}, we can form a pair of ordinary differential equations:
<math display="block" alt="d square big X over d x squared equals negative of omega over c quantity squared times big X, and d squared big T over d t squared equals negative omega squared times T">\frac{d^2}{dx^2}X=-\frac{\omega^2}{c^2}X, \qquad \frac{d^2}{dt^2}T = -\omega^2 T.</math>

Each of these is an eigenvalue equation with eigenvalues <math display="inline">-\frac{\omega^2}{c^2}</math> and {{math|−''ω''<sup>2</sup>}}, respectively. For any values of {{mvar|ω}} and {{mvar|c}}, the equations are satisfied by the functions
<math display="block">X(x) = \sin\left(\frac{\omega x}{c} + \varphi\right), \qquad T(t) = \sin(\omega t + \psi),</math>
where the phase angles {{mvar|φ}} and {{mvar|ψ}} are arbitrary real constants.

If we impose boundary conditions, for example that the ends of the string are fixed at {{math|1=''x'' = 0}} and {{math|1=''x'' = ''L''}}, namely {{math|1=''X''(0) = ''X''(''L'') = 0}}, and that {{math|1=''T''(0) = 0}}, we constrain the eigenvalues. For these boundary conditions, {{math|1=sin(''φ'') = 0}} and {{math|1=sin(''ψ'') = 0}}, so the phase angles {{math|1=''φ'' = ''ψ'' = 0}}, and
<math display="block" alt="sine of omega divided by c quantity equals 0">\sin\left(\frac{\omega L}{c}\right) = 0.</math>

This last boundary condition constrains {{mvar|ω}} to take a value {{math|1=''ω<sub>n</sub>'' = {{sfrac|''ncπ''|''L''}}}}, where {{mvar|n}} is any integer. Thus, the clamped string supports a family of standing waves of the form
<math display="block">h(x,t) = \sin\left(\frac{n\pi x}{L} \right) \sin(\omega_n t).</math>

In the example of a string instrument, the frequency {{math|''ω<sub>n</sub>''}} is the frequency of the {{mvar|n}}-th [[harmonic]], which is called the {{math|(''n'' − 1)}}-th [[overtone]].

===Schrödinger equation===
In [[quantum mechanics]], the [[Schrödinger equation]]
<math display="block">i \hbar \frac{\partial}{\partial t}\Psi(\mathbf{r},t) = H \Psi(\mathbf{r},t)</math>
with the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]]
<math display="block"> H = -\frac{\hbar^2}{2m}\nabla^2+ V(\mathbf{r},t)</math>
can be solved by separation of variables if the Hamiltonian does not depend explicitly on time.{{sfn|Davydov|1976|p=51}} In that case, the [[wave function]] {{math|1=Ψ('''r''',''t'') = ''φ''('''r''')''T''(''t'')}} leads to the two differential equations,
{{NumBlk||<math display="block"> H\varphi(\mathbf{r}) = E\varphi(\mathbf{r}),</math>|{{EquationRef|2}}}}
{{NumBlk||<math display="block"> i\hbar \frac{\partial T(t)}{\partial t} = ET(t).</math>|{{EquationRef|3}}}}
Both of these differential equations are eigenvalue equations with eigenvalue {{mvar|E}}. As shown in an earlier example, the solution of Equation {{EqNote|3}} is the exponential
<math display="block"> T(t) = e^{{-iEt}/{\hbar}}.</math>

Equation {{EqNote|2}} is the time-independent Schrödinger equation. The eigenfunctions {{mvar|φ<sub>k</sub>}} of the Hamiltonian operator are [[stationary state]]s of the quantum mechanical system, each with a corresponding energy {{mvar|E<sub>k</sub>}}. They represent allowable energy states of the system and may be constrained by boundary conditions.

The Hamiltonian operator {{mvar|H}} 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 oscillatory {{math|''T''(''t'')}},{{sfn|Davydov|1976|p=52}} <math display="inline"> \Psi(\mathbf{r},t) = \sum_k c_k \varphi_k(\mathbf{r}) e^{{-iE_kt}/{\hbar}} </math> or, for a system with a continuous spectrum,
<math display="block"> \Psi(\mathbf{r},t) = \int dE \, c_E \varphi_E(\mathbf{r}) e^{{-iEt}/{\hbar}}.</math>

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

===Signals and systems===
In the study of [[LTI system theory|signals and systems]], an eigenfunction of a system is a signal {{math|''f''(''t'')}} that, when input into the system, produces a response {{math|1=''y''(''t'') = ''λf''(''t'')}}, where {{mvar|λ}} is a complex scalar eigenvalue.{{sfn|Girod|Rabenstein|Stenger|2001|p=49}}

==See also==
* [[Eigenvalues and eigenvectors]]
* [[Hilbert–Schmidt theorem]]
* [[Spectral theory of ordinary differential equations]]
* [[Fixed point combinator]]
* [[Fourier transform#Eigenfunctions|Fourier transform eigenfunctions]]

==Notes==
{{notelist}}

===Citations===
{{Reflist|20em}}

==Works cited==
{{refbegin}}
*{{Cite book| title = Methods of Mathematical Physics
 | last1 = Courant | first1 = Richard
 | last2 = Hilbert | first2 = David
 | date = 1989 | publisher = Wiley
 | volume = 1
 | isbn = 047150447-5
}} (Volume 2: {{isbn| 047150439-4}})
*{{cite book| title = Quantum Mechanics | edition = 2nd
 | last = Davydov | first = A. S. | year = 1976
 | others = Translated, edited, and with additions by D. ter Haar
 | publisher = Pergamon Press | location = Oxford
 | isbn = 008020438-4
}}
*{{cite book| title = Signals and systems | edition = 2nd
 | last1 = Girod | first1 = Bernd
 | last2 = Rabenstein | first2 = Rudolf
 | last3 = Stenger | first3 = Alexander
 | author1-link = Bernd Girod
 | year = 2001
 | publisher = Wiley
 | isbn = 047198800-6
}}
*{{cite book| title = Mathematical Physics
 | last1 = Kusse | first1 = Bruce
 | last2 = Westwig | first2 = Erik
 | year = 1998
 | publisher = Wiley Interscience | location = New York
 | isbn = 047115431-8
}}
*{{cite web| title = Eigenfunction
 | last = Wasserman | first = Eric W. | year = 2016
 | website = MathWorld
 | publisher = [[Wolfram Research]]
 | url = http://mathworld.wolfram.com/Eigenfunction.html
 | access-date = April 12, 2016
}}
{{refend}}

==External links==
* More images (non-GPL) at [https://daugerresearch.com/orbitals/index.shtml Atom in a Box]

[[Category:Functional analysis]]