Editing Eigenfunction (section)

===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.