Editing Eigenfunction (section)

==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.