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Eigenfunction

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In <a href="Mathematics" title="Mathematics">mathematics</a>, an eigenfunction of a <a href="Linear_map" title="Linear map">linear operator</a> D defined on some <a href="Function_space" title="Function space">function space</a> is any non-zero <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> in that space that, when acted upon by D, is only multiplied by some scaling factor called an <a href="Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalue</a>. As an equation, this condition can be written as

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<annotation encoding="application/x-tex">{\displaystyle Df=\lambda f}</annotation>

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for some <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> eigenvalue <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda .}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda .}</annotation>

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</math><img src="./4bb9c58e3f6b2de892e10ef516f96f07da0423e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.002ex; height:2.176ex;" alt="{\displaystyle \lambda .}" loading="lazy"><a href="#cite_note-FOOTNOTEDavydov197620-1">[1]</a><a href="#cite_note-FOOTNOTEKusseWestwig1998435-2">[2]</a><a href="#cite_note-FOOTNOTEWasserman2016-3">[3]</a> The solutions to this equation may also be subject to <a href="Boundary_value_problem#boundary_value_conditions" title="Boundary value problem">boundary conditions</a> that limit the allowable eigenvalues and eigenfunctions.

An eigenfunction is a type of <a href="Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvector</a>.

<div class="mw-heading mw-heading2"><h2 id="Eigenfunctions">Eigenfunctions</h2></div>

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

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</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Df=\lambda f,}">

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<annotation encoding="application/x-tex">{\displaystyle Df=\lambda f,}</annotation>

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where λ is a scalar.<a href="#cite_note-FOOTNOTEDavydov197620-1">[1]</a><a href="#cite_note-FOOTNOTEKusseWestwig1998435-2">[2]</a><a href="#cite_note-FOOTNOTEWasserman2016-3">[3]</a> The solutions to Equation (<a href="#math_1">1</a>) 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 λ1, λ2, … or to a continuous set over some range. The set of all possible eigenvalues of D is sometimes called its <a href="Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">spectrum</a>, which may be discrete, continuous, or a combination of both.<a href="#cite_note-FOOTNOTEDavydov197620-1">[1]</a>

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 <a href="Degenerate_energy_levels#mathematics" title="Degenerate energy levels">degenerate</a> and the maximum number of linearly independent eigenfunctions associated with the same eigenvalue is the eigenvalue's degree of degeneracy or <a href="Eigenvalues_and_eigenvectors#Eigenspaces,_geometric_multiplicity,_and_the_eigenbasis" title="Eigenvalues and eigenvectors">geometric multiplicity</a>.<a href="#cite_note-FOOTNOTEDavydov197621-4">[4]</a><a href="#cite_note-FOOTNOTEKusseWestwig1998437-5">[5]</a>

<div class="mw-heading mw-heading3"><h3 id="Derivative_example">Derivative example</h3></div>

A widely used class of linear operators acting on infinite dimensional spaces are differential operators on the space C∞ of infinitely differentiable real or complex functions of a real or complex argument t. For example, consider the derivative operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {d}{dt}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}f(t)=\lambda f(t).}</annotation>

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This differential equation can be solved by multiplying both sides by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {dt}{f(t)}}}">

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<annotation encoding="application/x-tex">{\displaystyle f(t)=f_{0}e^{\lambda t},}</annotation>

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is the eigenfunction of the derivative operator, where f0 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left.{\frac {df}{dt}}\right|_{t=0}=2}">

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<annotation encoding="application/x-tex">{\textstyle \left.{\frac {df}{dt}}\right|_{t=0}=2}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle f(t)=e^{2t},}</annotation>

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

<div class="mw-heading mw-heading3"><h3 id="Link_to_eigenvalues_and_eigenvectors_of_matrices">Link to eigenvalues and eigenvectors of matrices</h3></div>

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 <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a> in the function space on which D is defined as

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<annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{\Omega }\ f^{*}(t)g(t)dt,}</annotation>

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integrated over some range of interest for t called Ω. The * denotes the <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a>.

Suppose the function space has an <a href="Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> given by the set of functions {u1(t), u2(t), …, un(t)}, where n may be infinite. For the orthonormal basis,

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<annotation encoding="application/x-tex">{\displaystyle \langle u_{i},u_{j}\rangle =\int _{\Omega }\ u_{i}^{*}(t)u_{j}(t)dt=\delta _{ij}={\begin{cases}1&i=j\\0&i\neq j\end{cases}},}</annotation>

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where δij is the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a> and can be thought of as the elements of the <a href="Identity_matrix" title="Identity matrix">identity matrix</a>.

Functions can be written as a linear combination of the basis functions,

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<annotation encoding="application/x-tex">{\displaystyle f(t)=\sum _{j=1}^{n}b_{j}u_{j}(t),}</annotation>

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for example through a <a href="Fourier_series" title="Fourier series">Fourier expansion</a> of f(t). The coefficients bj can be stacked into an n by 1 column vector b = [b1 b2 … bn]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 operator D with elements

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<annotation encoding="application/x-tex">{\displaystyle A_{ij}=\langle u_{i},Du_{j}\rangle =\int _{\Omega }\ u_{i}^{*}(t)Du_{j}(t)dt.}</annotation>

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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),

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<annotation encoding="application/x-tex">{\displaystyle Df(t)=\sum _{j=1}^{n}c_{j}u_{j}(t)=\sum _{j=1}^{n}b_{j}Du_{j}(t).}</annotation>

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation>

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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.

<div class="mw-heading mw-heading3"><h3 id="Eigenvalues_and_eigenfunctions_of_Hermitian_operators">Eigenvalues and eigenfunctions of Hermitian operators</h3></div>

Many of the operators encountered in physics are <a href="Self-adjoint_operator" title="Self-adjoint operator">Hermitian</a>. Suppose the linear operator D acts on a function space that is a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> with an orthonormal basis given by the set of functions {u1(t), u2(t), …, un(t)}, where n may be infinite. In this basis, the operator D has a matrix representation A with elements

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<annotation encoding="application/x-tex">{\displaystyle A_{ij}=\langle u_{i},Du_{j}\rangle =\int _{\Omega }dt\ u_{i}^{*}(t)Du_{j}(t).}</annotation>

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integrated over some range of interest for t denoted Ω.

By analogy with <a href="Hermitian_matrix" title="Hermitian matrix">Hermitian matrices</a>, D is a Hermitian operator if Aij = Aji*, or:<a href="#cite_note-FOOTNOTEKusseWestwig1998436-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation>

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Consider the Hermitian operator D with eigenvalues λ1, λ2, ... and corresponding eigenfunctions f1(t), f2(t), …. This Hermitian operator has the following properties:

<ul><li>Its eigenvalues are real, λi = λi*<a href="#cite_note-FOOTNOTEDavydov197621-4">[4]</a><a href="#cite_note-FOOTNOTEKusseWestwig1998436-6">[6]</a></li>

<li>Its eigenfunctions obey an orthogonality condition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f_{i},f_{j}\rangle =0}">

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<annotation encoding="application/x-tex">{\displaystyle \langle f_{i},f_{j}\rangle =0}</annotation>

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</math><img src="./cfcbc23026036780825561e618bc98a635b5d979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.092ex; height:3.009ex;" alt="{\displaystyle \langle f_{i},f_{j}\rangle =0}" loading="lazy"> if i ≠ j<a href="#cite_note-FOOTNOTEKusseWestwig1998436-6">[6]</a><a href="#cite_note-FOOTNOTEDavydov197624-7">[7]</a><a href="#cite_note-FOOTNOTEDavydov197629-8">[8]</a></li></ul>

The second condition always holds for λi ≠ λj. For degenerate eigenfunctions with the same eigenvalue λi, orthogonal eigenfunctions can always be chosen that span the eigenspace associated with λi, for example by using the <a href="Gram-Schmidt_process" class="mw-redirect" title="Gram-Schmidt process">Gram-Schmidt process</a>.<a href="#cite_note-FOOTNOTEKusseWestwig1998437-5">[5]</a> 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 <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>, respectively.<a href="#cite_note-FOOTNOTEDavydov197629-8">[8]</a><a href="#cite_note-FOOTNOTEDavydov197625-9">[9]</a>

For many Hermitian operators, notably <a href="Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville operators</a>, a third property is

<ul><li>Its eigenfunctions form a basis of the function space on which the operator is defined<a href="#cite_note-FOOTNOTEKusseWestwig1998437-5">[5]</a></li></ul>

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.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Vibrating_strings">Vibrating strings</h3></div>

Let h(x, t) denote the transverse displacement of a stressed elastic chord, such as the <a href="Vibrating_string" class="mw-redirect" title="Vibrating string">vibrating strings</a> of a <a href="String_instrument" title="String instrument">string instrument</a>, as a function of the position x along the string and of time t. Applying the laws of mechanics to <a href="Infinitesimal" title="Infinitesimal">infinitesimal</a> portions of the string, the function h satisfies the <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}h}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}h}{\partial x^{2}}},}</annotation>

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which is called the (one-dimensional) <a href="Wave_equation" title="Wave equation">wave equation</a>. Here c is a constant speed that depends on the tension and mass of the string.

This problem is amenable to the method of <a href="Separation_of_variables" title="Separation of variables">separation of variables</a>. 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:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}}{dx^{2}}}X=-{\frac {\omega ^{2}}{c^{2}}}X,\qquad {\frac {d^{2}}{dt^{2}}}T=-\omega ^{2}T.}</annotation>

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Each of these is an eigenvalue equation with eigenvalues

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<annotation encoding="application/x-tex">{\textstyle -{\frac {\omega ^{2}}{c^{2}}}}</annotation>

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</math><img src="./3c8eddb4e00baa2243173304a7b19c54099d2cbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:4.498ex; height:4.343ex;" alt="{\textstyle -{\frac {\omega ^{2}}{c^{2}}}}" loading="lazy"> and −ω2, respectively. For any values of ω and c, the equations are satisfied by the functions

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<annotation encoding="application/x-tex">{\displaystyle X(x)=\sin \left({\frac {\omega x}{c}}+\varphi \right),\qquad T(t)=\sin(\omega t+\psi ),}</annotation>

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where the phase angles φ and ψ are arbitrary real constants.

If we impose boundary conditions, for example that the ends of the string are fixed at x = 0 and x = L, namely X(0) = X(L) = 0, and that T(0) = 0, we constrain the eigenvalues. For these boundary conditions, sin(φ) = 0 and sin(ψ) = 0, so the phase angles φ = ψ = 0, and

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<annotation encoding="application/x-tex">{\displaystyle \sin \left({\frac {\omega L}{c}}\right)=0.}</annotation>

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This last boundary condition constrains ω to take a value ωn = <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠ncπ/L⁠, where n is any integer. Thus, the clamped string supports a family of standing waves of the form

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<annotation encoding="application/x-tex">{\displaystyle h(x,t)=\sin \left({\frac {n\pi x}{L}}\right)\sin(\omega _{n}t).}</annotation>

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In the example of a string instrument, the frequency ωn is the frequency of the n-th <a href="Harmonic" title="Harmonic">harmonic</a>, which is called the (n − 1)-th <a href="Overtone" title="Overtone">overtone</a>.

<div class="mw-heading mw-heading3"><h3 id="Schrödinger_equation">Schrödinger equation</h3></div>

In <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>

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<annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=H\Psi (\mathbf {r} ,t)}</annotation>

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with the <a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian operator</a>

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<annotation encoding="application/x-tex">{\displaystyle H=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)}</annotation>

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can be solved by separation of variables if the Hamiltonian does not depend explicitly on time.<a href="#cite_note-FOOTNOTEDavydov197651-10">[10]</a> In that case, the <a href="Wave_function" title="Wave function">wave function</a> Ψ(r,t) = φ(r)T(t) leads to the two differential equations,

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<annotation encoding="application/x-tex">{\displaystyle H\varphi (\mathbf {r} )=E\varphi (\mathbf {r} ),}</annotation>

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</math></td> <td></td> <td class="nowrap">2</td></tr></tbody></table>

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<annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial T(t)}{\partial t}}=ET(t).}</annotation>

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</math></td> <td></td> <td class="nowrap">3</td></tr></tbody></table>

Both of these differential equations are eigenvalue equations with eigenvalue E. As shown in an earlier example, the solution of Equation (<a href="#math_3">3</a>) is the exponential

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<annotation encoding="application/x-tex">{\displaystyle T(t)=e^{{-iEt}/{\hbar }}.}</annotation>

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Equation (<a href="#math_2">2</a>) is the time-independent Schrödinger equation. The eigenfunctions φk of the Hamiltonian operator are <a href="Stationary_state" title="Stationary state">stationary states</a> of the quantum mechanical system, each with a corresponding energy Ek. They represent allowable energy states of the system and may be constrained by boundary conditions.

The Hamiltonian operator 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 T(t),<a href="#cite_note-FOOTNOTEDavydov197652-11">[11]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Psi (\mathbf {r} ,t)=\sum _{k}c_{k}\varphi _{k}(\mathbf {r} )e^{{-iE_{k}t}/{\hbar }}}">

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<annotation encoding="application/x-tex">{\textstyle \Psi (\mathbf {r} ,t)=\sum _{k}c_{k}\varphi _{k}(\mathbf {r} )e^{{-iE_{k}t}/{\hbar }}}</annotation>

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</math><img src="./1cca1486af3e7bb06927ffe5d28df8e6bf0bb7b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.811ex; height:3.343ex;" alt="{\textstyle \Psi (\mathbf {r} ,t)=\sum _{k}c_{k}\varphi _{k}(\mathbf {r} )e^{{-iE_{k}t}/{\hbar }}}" loading="lazy"> or, for a system with a continuous spectrum,

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<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} ,t)=\int dE\,c_{E}\varphi _{E}(\mathbf {r} )e^{{-iEt}/{\hbar }}.}</annotation>

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

<div class="mw-heading mw-heading3"><h3 id="Signals_and_systems">Signals and systems</h3></div>

In the study of <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">signals and systems</a>, an eigenfunction of a system is a signal f(t) that, when input into the system, produces a response y(t) = λf(t), where λ is a complex scalar eigenvalue.<a href="#cite_note-FOOTNOTEGirodRabensteinStenger200149-12">[12]</a>

<ul><li><a href="Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li>

<li><a href="Fixed_point_combinator" class="mw-redirect" title="Fixed point combinator">Fixed point combinator</a></li>

<li><a href="Fourier_transform#Eigenfunctions" title="Fourier transform">Fourier transform eigenfunctions</a></li>

<li><a href="Hilbert%E2%80%93Schmidt_theorem" title="Hilbert–Schmidt theorem">Hilbert–Schmidt theorem</a></li>

<li><a href="Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2></div>

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</style><div class="reflist reflist-columns references-column-width" style="column-width: 20em;">

<li id="cite_note-FOOTNOTEDavydov197620-1">^ <a href="#cite_ref-FOOTNOTEDavydov197620_1-0">a</a> <a href="#cite_ref-FOOTNOTEDavydov197620_1-1">b</a> <a href="#cite_ref-FOOTNOTEDavydov197620_1-2">c</a> <a href="#CITEREFDavydov1976">Davydov 1976</a>, p. 20.

</li>

<li id="cite_note-FOOTNOTEKusseWestwig1998435-2">^ <a href="#cite_ref-FOOTNOTEKusseWestwig1998435_2-0">a</a> <a href="#cite_ref-FOOTNOTEKusseWestwig1998435_2-1">b</a> <a href="#CITEREFKusseWestwig1998">Kusse & Westwig 1998</a>, p. 435.

</li>

<li id="cite_note-FOOTNOTEWasserman2016-3">^ <a href="#cite_ref-FOOTNOTEWasserman2016_3-0">a</a> <a href="#cite_ref-FOOTNOTEWasserman2016_3-1">b</a> <a href="#CITEREFWasserman2016">Wasserman 2016</a>.

</li>

<li id="cite_note-FOOTNOTEDavydov197621-4">^ <a href="#cite_ref-FOOTNOTEDavydov197621_4-0">a</a> <a href="#cite_ref-FOOTNOTEDavydov197621_4-1">b</a> <a href="#CITEREFDavydov1976">Davydov 1976</a>, p. 21.

</li>

<li id="cite_note-FOOTNOTEKusseWestwig1998437-5">^ <a href="#cite_ref-FOOTNOTEKusseWestwig1998437_5-0">a</a> <a href="#cite_ref-FOOTNOTEKusseWestwig1998437_5-1">b</a> <a href="#cite_ref-FOOTNOTEKusseWestwig1998437_5-2">c</a> <a href="#CITEREFKusseWestwig1998">Kusse & Westwig 1998</a>, p. 437.

</li>

<li id="cite_note-FOOTNOTEKusseWestwig1998436-6">^ <a href="#cite_ref-FOOTNOTEKusseWestwig1998436_6-0">a</a> <a href="#cite_ref-FOOTNOTEKusseWestwig1998436_6-1">b</a> <a href="#cite_ref-FOOTNOTEKusseWestwig1998436_6-2">c</a> <a href="#CITEREFKusseWestwig1998">Kusse & Westwig 1998</a>, p. 436.

</li>

<li id="cite_note-FOOTNOTEDavydov197624-7"><a href="#cite_ref-FOOTNOTEDavydov197624_7-0">^</a> <a href="#CITEREFDavydov1976">Davydov 1976</a>, p. 24.

</li>

<li id="cite_note-FOOTNOTEDavydov197629-8">^ <a href="#cite_ref-FOOTNOTEDavydov197629_8-0">a</a> <a href="#cite_ref-FOOTNOTEDavydov197629_8-1">b</a> <a href="#CITEREFDavydov1976">Davydov 1976</a>, p. 29.

</li>

<li id="cite_note-FOOTNOTEDavydov197625-9"><a href="#cite_ref-FOOTNOTEDavydov197625_9-0">^</a> <a href="#CITEREFDavydov1976">Davydov 1976</a>, p. 25.

</li>

<li id="cite_note-FOOTNOTEDavydov197651-10"><a href="#cite_ref-FOOTNOTEDavydov197651_10-0">^</a> <a href="#CITEREFDavydov1976">Davydov 1976</a>, p. 51.

</li>

<li id="cite_note-FOOTNOTEDavydov197652-11"><a href="#cite_ref-FOOTNOTEDavydov197652_11-0">^</a> <a href="#CITEREFDavydov1976">Davydov 1976</a>, p. 52.

</li>

<li id="cite_note-FOOTNOTEGirodRabensteinStenger200149-12"><a href="#cite_ref-FOOTNOTEGirodRabensteinStenger200149_12-0">^</a> <a href="#CITEREFGirodRabensteinStenger2001">Girod, Rabenstein & Stenger 2001</a>, p. 49.

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Works_cited">Works cited</h2></div>

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</style><cite id="CITEREFCourantHilbert1989" class="citation book cs1">Courant, Richard; Hilbert, David (1989). Methods of Mathematical Physics. Vol. 1. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>047150447-5</bdi>.</cite> (Volume 2: <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>047150439-4</bdi>.)</li>

<li><cite id="CITEREFDavydov1976" class="citation book cs1">Davydov, A. S. (1976). Quantum Mechanics. Translated, edited, and with additions by D. ter Haar (2nd ed.). Oxford: Pergamon Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>008020438-4</bdi>.</cite></li>

<li><cite id="CITEREFGirodRabensteinStenger2001" class="citation book cs1"><a href="Bernd_Girod" title="Bernd Girod">Girod, Bernd</a>; Rabenstein, Rudolf; Stenger, Alexander (2001). Signals and systems (2nd ed.). Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>047198800-6</bdi>.</cite></li>

<li><cite id="CITEREFKusseWestwig1998" class="citation book cs1">Kusse, Bruce; Westwig, Erik (1998). Mathematical Physics. New York: Wiley Interscience. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>047115431-8</bdi>.</cite></li>

<li><cite id="CITEREFWasserman2016" class="citation web cs1">Wasserman, Eric W. (2016). <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Eigenfunction.html">"Eigenfunction"</a>. MathWorld. <a href="Wolfram_Research" title="Wolfram Research">Wolfram Research</a>. Retrieved April 12, 2016.</cite></li></ul>

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