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In [[mathematics]] and its applications, a classical '''Sturm–Liouville equation''', named after [[Jacques Charles François Sturm]] (1803–1855) and [[Joseph Liouville]] (1809–1882), is a real second-order linear [[differential equation]] of the form:
<div style="text-align: right;">
<div style="float: left;"><math> -\frac{d}{dx}\left[p(x)\frac{dy}{ dx}\right]+q(x)y=\lambda w(x)y.</math></div>
<span id="(1)">(1)</span>
</div>
where ''y'' is a function of the free variable ''x''. Here the functions ''p''(''x'')&nbsp;>&nbsp;0 has a ''continuous derivative'', ''q''(''x''), and ''w''(''x'')&nbsp;>&nbsp;0 are specified at the outset, and in the simplest of cases are continuous on the finite closed interval [''a'',''b'']. In addition, the function ''y'' is typically required to satisfy some [[boundary condition]]s at ''a'' and ''b''. The function ''w''(''x''), which is sometimes called ''r''(''x''), is called the "weight" or "density" function.
The value of &lambda; is not specified in the equation; finding the values of &lambda;  for which there exists a non-trivial solution of [[#(1) | (1)]] satisfying the boundary conditions is part of the problem called the '''Sturm–Liouville problem''' (S–L).
Such values of &lambda; when they exist are called the [[eigenvalues]] of the boundary value problem defined by [[#(1) | (1)]] and the prescribed set of boundary conditions. The corresponding solutions (for such a &lambda;) are the [[eigenfunction]]s of this problem. Under normal assumptions on the coefficient functions ''p''(''x''), ''q''(''x''), and ''w''(''x'') above, they induce a [[hermitian operator|Hermitian]] [[differential operator]] in some [[function space]] defined by [[boundary value problem|boundary conditions]]. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their [[Complete metric space|completeness]] in a suitable [[function space]] became known as '''Sturm–Liouville theory'''. This theory is important in applied mathematics, where S–L problems  occur very commonly, particularly when dealing with linear [[partial differential equation]]s that are [[separation of variables|separable]].
== Sturm–Liouville theory ==
Under the assumptions that the S–L problem is regular, that is, ''p''(''x'')<sup>&minus;1</sup>&nbsp;>&nbsp;0, ''q''(''x''), and ''w''(''x'')&nbsp;>&nbsp;0 are real-valued [[Lebesgue integrable|integrable]] functions over the finite interval [''a'',&nbsp;''b''],
with ''separated boundary conditions'' of the form:
<div style="text-align: right;">
<div style="float: left;"><math> y(a)\cos \alpha - p(a)y^\prime (a)\sin \alpha = 0,</math></div>
<span id="(2)">(2)</span>
</div>
<div style="text-align: right;">
<div style="float: left;"><math> y(b)\cos \beta - p(b)y^\prime (b)\sin \beta = 0,</math></div>
<span id="(3)">(3)</span>
</div>
where <math>\alpha, \beta \in [0, \pi),</math>
the main tenet of '''Sturm–Liouville theory''' states that:
* The eigenvalues &lambda;<sub>1</sub>, &lambda;<sub>2</sub>, &lambda;<sub>3</sub>, ... of the regular Sturm–Liouville problem [[#(1) | (1)]] -[[#(2) | (2)]] -[[#(3) | (3)]] are real and can be ordered such that:
:: <math>\lambda_1 < \lambda_2 < \lambda_3 < \cdots < \lambda_n < \cdots \to \infty; \, </math>
* Corresponding to each eigenvalue &lambda;<sub>''n''</sub> is a unique (up to a normalization constant) eigenfunction ''y''<sub>''n''</sub>(''x'') which has exactly ''n''&nbsp;&minus;&nbsp;1 zeros in (''a'',&nbsp;''b''). The eigenfunction ''y''<sub>''n''</sub>(''x'') is called the ''n''-th ''fundamental solution'' satisfying the regular Sturm–Liouville problem [[#(1) | (1)]] -[[#(2) | (2)]] -[[#(3) | (3)]].
* The normalized eigenfunctions form an [[orthonormal basis]] 
::<math> \int_a^b y_n(x)y_m(x)w(x)\,dx = \delta_{mn},</math> 
 
:in the [[Hilbert space]] [[Lebesgue space|''L''<sup>2</sup>([''a'',&nbsp;''b''],''w''(''x'')&nbsp;''dx'')]]. Here &delta;<sub>''mn''</sub> is a [[Kronecker delta]].
Note that, unless  ''p''(''x'') is continuously differentiable and ''q''(''x''), ''w''(''x'') are continuous, the equation has to be understood in a [[weak solution|weak sense]].
== Sturm–Liouville form ==
The differential equation [[#(1) | (1)]] is said to be in '''Sturm–Liouville form''' or '''self-adjoint form'''. All second-order linear [[ordinary differential equation]]s can be recast in the form on the left-hand side of [[#(1) | (1)]] by multiplying both sides of the equation by an appropriate [[integrating factor]] (although the same is not true of second-order [[partial differential equation]]s, or if ''y'' is a vector.)
=== Examples ===
The [[Bessel equation]]:
: <math>x^2y''+xy'+(\lambda^2x^2-\nu^2)y=0\,</math>
can be written in Sturm–Liouville form as:
: <math>(xy')'+(\lambda^2 x-\nu^2/x)y=0.\,</math>
The [[Legendre polynomials|Legendre equation]]:
: <math>(1-x^2)y''-2xy'+\nu(\nu+1)y=0\;\!</math>
can easily be put into Sturm–Liouville form, since ''D''(1&nbsp;&minus;&nbsp;''x''<sup>2</sup>) = &minus;2''x'', so, the Legendre equation is equivalent to:
: <math>[(1-x^2)y']'+\nu(\nu+1)y=0\;\!</math>
Less simple is such a differential equation as:
: <math>x^3y''-xy'+2y=0.\,</math>
Divide throughout by ''x''<sup>3</sup>:
: <math>y''-{x\over x^3}y'+{2\over x^3}y=0</math>
Multiplying throughout by an [[integrating factor]] of:
: <math>e^{\int -{x / x^3}\,dx}=e^{\int -{1 / x^2}\, dx}=e^{1 / x},</math>
gives:
: <math>e^{1 / x}y''-{e^{1 / x} \over x^2} y'+ {2 e^{1 / x} \over x^3} y = 0</math>
which can be easily put into Sturm–Liouville form since:
: <math>D e^{1 / x} = -{e^{1 / x} \over x^2} </math>
so the differential equation is equivalent to:
: <math>(e^{1 / x}y')'+{2 e^{1 / x} \over x^3} y =0.</math>
In general, given a differential equation:
: <math>P(x)y''+Q(x)y'+R(x)y=0\,</math>
dividing by ''P''(''x''), multiplying through by the integrating factor: 
: <math>e^{\int {Q(x) / P(x)}\,dx},</math>
and then collecting gives the Sturm–Liouville form.
== Sturm–Liouville equations as self-adjoint differential operators ==
The map:
: <math>L  u  = {1 \over w(x)} \left(-{d\over dx}\left[p(x){du\over dx}\right]+q(x)u \right)</math>
can be viewed as a [[linear operator]] mapping a function ''u'' to another function ''Lu''. We may study this linear operator in the context of [[functional analysis]]. In fact, equation [[#(1) | (1)]] can be written as:
: <math>L  u  = \lambda u \,.</math>
This is precisely the [[eigenvalue]] problem; that is, we are trying to find the eigenvalues &lambda;<sub>1</sub>, &lambda;<sub>2</sub>, &lambda;<sub>3</sub>, ... and the corresponding eigenvectors ''u''<sub>1</sub>, ''u''<sub>2</sub>, ''u''<sub>3</sub>, ... of the ''L'' operator. The proper setting for this problem is the [[Hilbert space]] [[Lp space#Weighted Lp spaces|''L''<sup>2</sup>([''a'',&nbsp;''b''],''w''(''x'')&nbsp;''dx'')]] with scalar product:
: <math> \langle f, g\rangle = \int_{a}^{b} \overline{f(x)} g(x)w(x)\,dx.</math> 
In this space ''L'' is defined on sufficiently smooth functions which satisfy the above [[boundary value problem|boundary condition]]s. Moreover, ''L'' gives rise to a [[self-adjoint]] operator.
This can be seen formally by using [[integration by parts]] twice, where the boundary terms vanish by virtue of the boundary conditions. It then follows that the eigenvalues of a Sturm–Liouville operator are real and that eigenfunctions of ''L'' corresponding to different eigenvalues are orthogonal. However, this operator is unbounded and hence existence of an orthonormal basis of eigenfunctions is not evident. To overcome this problem one looks at the [[resolvent]]:
::<math> (L - z)^{-1}, \qquad z \in\mathbb{C},</math>
where ''z'' is chosen to be some real number which is not an eigenvalue. Then, computing the resolvent amounts to solving the inhomogeneous equation, which can be done using the [[variation of parameters]] formula. This shows that the resolvent is an [[integral operator]] with a continuous symmetric kernel (the [[Green's function]] of the problem). As a consequence of the [[Arzelà–Ascoli theorem]] this integral operator is compact and existence of a sequence of eigenvalues &alpha;<sub>''n''</sub> which converge to 0 and eigenfunctions which form an orthonormal basis follows from the [[compact operator on Hilbert space|spectral theorem for compact operators]]. Finally, note that <math>(L-z)^{-1} u = \alpha u</math> is equivalent to <math>L u = (z+\alpha^{-1}) u</math>.
If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls ''L'' singular. In this case the spectrum does no longer consist of eigenvalues alone and can contain a continuous component. There is still an associated eigenfunction expansion (similar to Fourier series versus Fourier transform). This is important in [[quantum mechanics]], since the one-dimensional [[Schrödinger equation]] is a special case of a S–L equation.
== Example ==
We wish to find a function ''u''(''x'') which solves the following Sturm–Liouville problem:
<div style="text-align: right;">
<div style="float: left;"><math> L  u  = \frac{d^2u}{dx^2} = \lambda u</math></div>
<span id="(4)">(4)</span>
</div>
where the unknowns are ''&lambda;'' and ''u''(''x''). As above, we must add boundary conditions, we take for example:
:<math> u(0) = u(\pi) = 0 \, </math>
Observe that if ''k'' is any integer, then the function:
:<math> u(x) = \sin kx \, </math>
is a solution with eigenvalue &lambda; = &minus;''k''<sup>2</sup>. We know that the solutions of a S–L problem form an orthogonal basis, and we know from Fourier series that this set of sinusoidal functions is an orthogonal basis. Since orthogonal bases are always maximal (by definition) we conclude that the S–L problem in this case has no other eigenvectors.
Given the preceding, let us now solve the inhomogeneous problem:
:<math>L  u  =x, \qquad x\in(0,\pi)</math>
with the same boundary conditions. In this case, we must write ''f''(''x'') = ''x'' in a Fourier series. The reader may check, either by integrating &int;exp(''ikx'')''x''&nbsp;d''x'' or by consulting a table of Fourier transforms, that we thus obtain:
:<math>L  u  =\sum_{k=1}^{\infty}-2\frac{(-1)^k}{k}\sin kx.</math>
This particular Fourier series is troublesome because of its poor convergence properties. It is not clear ''a priori'' whether the series converges pointwise. Because of Fourier analysis, since the Fourier coefficients are "square-summable", the Fourier series converges in ''L''<sup>2</sup> which is all we need for this particular theory to function. We mention for the interested reader that in this case we may rely on a result which says that Fourier's series converges at every point of differentiability, and at jump points (the function ''x'', considered as a periodic function, has a jump at &pi;) converges to the average of the left and right limits (see [[convergence of Fourier series]]).
Therefore, by using formula [[#(4) | (4)]], we obtain the solution:
:<math>u=\sum_{k=1}^{\infty}2\frac{(-1)^k}{k^3}\sin kx.</math>
In this case, we could have found the answer using antidifferentiation. This technique yields ''u'' =&nbsp;(''x''<sup>3</sup>&nbsp;&minus;&nbsp;''&pi;''<sup>2</sup>''x'')/6, whose Fourier series agrees with the solution we found. The antidifferentiation technique is no longer useful in most cases when the differential equation is in many variables.
== Application to normal modes ==
Suppose we are interested in the modes of vibration of a thin membrane, held in a rectangular frame, 0 < ''x'' < ''L''<sub>1</sub>, 0 < ''y'' < ''L''<sub>2</sub>. We know the equation of motion for the vertical membrane's displacement, ''W''(''x'', ''y'', ''t'') is given by the [[wave equation]]:
:<math>\frac{\partial^2W}{\partial x^2}+\frac{\partial^2W}{\partial y^2} = \frac{1}{c^2}\frac{\partial^2W}{\partial t^2}.</math>
The equation is [[separation of variables|separable]] (substituting ''W'' = ''X''(''x'') &times; ''Y''(''y'') &times; ''T''(''t'')), and the normal mode solutions that have [[harmonic]] time dependence and satisfy the boundary conditions ''W'' = 0 at ''x'' = 0, ''L''<sub>1</sub> and ''y'' = 0, ''L''<sub>2</sub> are given by:
:<math>W_{mn}(x,y,t) = A_{mn}\sin\left(\frac{m\pi x}{L_1}\right)\sin\left(\frac{n\pi y}{L_2}\right)\cos\left(\omega_{mn}t\right)</math>
where ''m'' and ''n'' are non-zero [[integer]]s, ''A<sub>mn</sub>'' is an arbitrary constant and:
: <math>\omega^2_{mn} = c^2 \left(\frac{m^2\pi^2}{L_1^2}+\frac{n^2\pi^2}{L_2^2}\right).</math>
Since the eigenfunctions ''W<sub>mn</sub>'' form a basis, an arbitrary initial displacement can be decomposed into a sum of these modes, which each vibrate at their individual frequencies <math>\omega_{mn}</math>. Infinite sums are also valid, as long as they [[convergence|converge]].
==See also==
* [[Normal mode]]
* [[Self-adjoint]]
== References ==
* P. Hartman, ''Ordinary Differential Equations'', SIAM, Philadelphia, 2002 (2nd edition). ISBN 978-0-898715-10-1
* A. D. Polyanin and V. F. Zaitsev, ''Handbook of Exact Solutions for Ordinary Differential Equations'', Chapman & Hall/CRC Press, Boca Raton, 2003 (2nd edition). ISBN 1-58488-297-2
* G. Teschl, ''Ordinary Differential Equations and Dynamical Systems'', http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ (Chapter 5)
* G. Teschl, ''Mathematical Methods in Quantum Mechanics and Applications to Schrödinger Operators'', http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ (see Chapter 9 for singular S-L operators and connections with quantum mechanics)
* A. Zettl, ''Sturm–Liouville Theory'', American Mathematical Society, 2005. ISBN 0-8218-3905-5.

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