Sturm–Liouville theory
In mathematics and its applications, a classical Sturm–Liouville theory, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is the theory of a real secondorder linear differential equation of the form

(1)
where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative. In this simplest of all cases, this function "y" is called a solution if it is continuously differentiable on (a,b) and satisfies the equation (1) at every point in (a,b). In addition, the unknown function y is typically required to satisfy some boundary conditions at a and b. The function w(x), which is sometimes called r(x), is called the "weight" or "density" function.
The value of λ is not specified in the equation; finding the values of λ for which there exists a nontrivial solution of (1) satisfying the boundary conditions is part of the problem called the Sturm–Liouville (S–L) problem.
Such values of λ, when they exist, are called the eigenvalues of the boundary value problem defined by (1) and the prescribed set of boundary conditions. The corresponding solutions (for such a λ) are the eigenfunctions of this problem. Under normal assumptions on the coefficient functions p(x), q(x), and w(x) above, they induce a Hermitian differential operator in some function space defined by boundary conditions. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their 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 equations that are separable.
A Sturm–Liouville (S–L) problem is said to be regular if p(x), w(x) > 0, and p(x), p'(x), q(x), and w(x) are continuous functions over the finite interval [a, b], and has separated boundary conditions of the form

(2)

(3)
Under the assumption that the S–L problem is regular, the main tenet of Sturm–Liouville theory states that:
 The eigenvalues λ_{1}, λ_{2}, λ_{3}, ... of the regular Sturm–Liouville problem (1)(2)(3) are real and can be ordered such that
 Corresponding to each eigenvalue λ_{n} is a unique (up to a normalization constant) eigenfunction y_{n}(x) which has exactly n − 1 zeros in (a, b). The eigenfunction y_{n}(x) is called the nth fundamental solution satisfying the regular Sturm–Liouville problem (1)(2)(3).
 The normalized eigenfunctions form an orthonormal basis
 in the Hilbert space L^{2}([a, b], w(x)dx). Here δ_{mn} is the 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 sense.
Sturm–Liouville form
The differential equation (1) is said to be in Sturm–Liouville form or selfadjoint form. All secondorder linear ordinary differential equations can be recast in the form on the lefthand side of (1) by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of secondorder partial differential equations, or if y is a vector.)
Examples
The Bessel equation
which can be written in Sturm–Liouville form as
The Legendre equation
which can easily be put into Sturm–Liouville form, since , so the Legendre equation is equivalent to
An example using an integrating factor
Divide throughout by x^{3}:
Multiplying throughout by an integrating factor of
gives
which can be easily put into Sturm–Liouville form since
so the differential equation is equivalent to
The integrating factor for a general second order differential equation
multiplying through by the integrating factor
and then collecting gives the Sturm–Liouville form:
or, explicitly,
Sturm–Liouville equations as selfadjoint differential operators
The map
can be viewed as a linear operator mapping a function u to another function Lu. One may study this linear operator in the context of functional analysis. In fact, equation (1) can be written as
This is precisely the eigenvalue problem; that is, one is trying to find the eigenvalues λ_{1}, λ_{2}, λ_{3}, ... and the corresponding eigenvectors u_{1}, u_{2}, u_{3}, ... of the L operator. The proper setting for this problem is the Hilbert space L^{2}([a, b], w(x) dx) with scalar product
In this space L is defined on sufficiently smooth functions which satisfy the above boundary conditions. Moreover, L gives rise to a selfadjoint 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
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 α_{n} which converge to 0 and eigenfunctions which form an orthonormal basis follows from the spectral theorem for compact operators. Finally, note that
are equivalent.
If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls L singular. In this case, the spectrum no longer consists 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 onedimensional timeindependent 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:

(4)
where the unknowns are λ and u(x). As above, we must add boundary conditions, we take for example
Observe that if k is any integer, then the function
is a solution with eigenvalue λ = k^{2}. 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
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 ∫e ^{ikx}x dx or by consulting a table of Fourier transforms, that we thus obtain
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 "squaresummable", the Fourier series converges in L^{2} 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 converge at every point of differentiability, and at jump points (the function x, considered as a periodic function, has a jump at π) converges to the average of the left and right limits (see convergence of Fourier series).
Therefore, by using formula (4), we obtain that the solution is
In this case, we could have found the answer using antidifferentiation. This technique yields
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
Certain partial differential equations can be solved with the help of S–L theory. Suppose we are interested in the modes of vibration of a thin membrane, held in a rectangular frame, 0 ≤ x ≤ L_{1}, 0 ≤ y ≤ L_{2}. The equation of motion for the vertical membrane's displacement, W(x, y, t) is given by the wave equation:
The method of separation of variables suggests looking first for solutions of the simple form W = X(x) × Y(y) × T(t). For such a function W the partial differential equation becomes X"/X + Y"/Y = (1/c^{2})T"/T. Since the three terms of this equation are functions of x,y,t separately, they must be constants. For example, the first term gives X" = λX for a constant λ. The boundary conditions ("held in a rectangular frame") are W = 0 when x = 0, L_{1} or y = 0, L_{2} and define the simplest possible S–L eigenvalue problems as in the example, yielding the "normal mode solutions" for W with harmonic time dependence,
where m and n are nonzero integers, A_{mn} are arbitrary constants, and
The functions W_{mn} form a basis for the Hilbert space of (generalized) solutions of the wave equation; that is, an arbitrary solution W can be decomposed into a sum of these modes, which vibrate at their individual frequencies . This representation may require a convergent infinite sum.
Representation of solutions and numerical calculation
The Sturm–Liouville differential equation (1) with boundary conditions may be solved in practice by a variety of numerical methods. In difficult cases, one may need to carry out the intermediate calculations to several hundred decimal places of accuracy in order to obtain the eigenvalues correctly to a few decimal places.
1. Shooting methods.^{[1]}^{[2]} These methods proceed by guessing a value of λ, solving an initial value problem defined by the boundary conditions at one endpoint, say, a, of the interval [a, b], comparing the value this solution takes at the other endpoint b with the other desired boundary condition, and finally increasing or decreasing λ as necessary to correct the original value. This strategy is not applicable for locating complex eigenvalues.
3. The Spectral Parameter Power Series (SPPS) method^{[3]} makes use of a generalization of the following fact about second order ordinary differential equations: if y is a solution which does not vanish at any point of [a,b], then the function
is a solution of the same equation and is linearly independent from y. Further, all solutions are linear combinations of these two solutions. In the SPPS algorithm, one must begin with an arbitrary value λ_{0}^{*} (often λ_{0}^{*} = 0; it does not need to be an eigenvalue) and any solution y_{0} of (1) with λ = λ_{0}^{*} which does not vanish on [a, b]. (Discussion below of ways to find appropriate y_{0} and λ_{0}^{*}.) Two sequences of functions X^{(n)}(t), X^{~(n)}(t) on [a, b], referred to as iterated integrals, are defined recursively as follows. First when n = 0, they are taken to be identically equal to 1 on [a, b]. To obtain the next functions they are multiplied alternately by 1/(py_{0}^{2}) and wy_{0}^{2} and integrated, specifically

(5)

(6)
when n > 0. The resulting iterated integrals are now applied as coefficients in the following two power series in λ:
Then for any λ (real or complex), u_{0} and u_{1} are linearly independent solutions of the corresponding equation (1). (The functions p(x) and q(x) take part in this construction through their influence on the choice of y_{0}.)
Next one chooses coefficients c_{0}, c_{1} so that the combination y = c_{0}u_{0} + c_{1}u_{1} satisfies the first boundary condition (2). This is simple to do since X^{(n)}(a) = 0 and X^{~(n)}(a) = 0, for n > 0. The values of X^{(n)}(b) and X^{~(n)}(b) provide the values of u_{0}(b) and u_{1}(b) and the derivatives u_{0}'(b) and u_{1}'(b), so the second boundary condition (3) becomes an equation in a power series in λ. For numerical work one may truncate this series to a finite number of terms, producing a calculable polynomial in λ whose roots are approximations of the soughtafter eigenvalues.
When λ = λ_{0}, this reduces to the original construction described above for a solution linearly independent to a given one. The representations (5),(6) also have theoretical applications in Sturm–Liouville theory.^{[3]}
Construction of a nonvanishing solution
The SPPS method can, itself, be used to find a starting solution y_{0}. Consider the equation (py')' = μqy; i.e., q, w, and λ are replaced in (1) by 0, −q, and μ respectively. Then the constant function 1 is a nonvanishing solution corresponding to the eigenvalue μ_{0} = 0. While there is no guarantee that u_{0} or u_{1} will not vanish, the complex function y_{0} = u_{0} + iu_{1} will never vanish because two linearly independent solutions of a regular S–L equation cannot vanish simultaneously as a consequence of the Sturm separation theorem. This trick gives a solution y_{0} of (1) for the value λ_{0} = 0. In practice if (1) has real coefficients, the solutions based on y_{0} will have very small imaginary parts which must be discarded.
Application to PDEs
For a linear second order in one spatial dimension and first order in time of the form:
Let us apply separation of variables, which in doing we must impose that:
Then our above PDE may be written as:
Where
Since, by definition, and are independent of time t and and are independent of position x, then both sides of the above equation must be equal to a constant:
The first of these equations must be solved as a Sturm–Liouville problem. Since there is no general analytic (exact) solution to Sturm–Liouville problems, we can assume we already have the solution to this problem, that is, we have the eigenfunctions and eigenvalues . The second of these equations can be analytically solved once the eigenvalues are known.
Where:
See also
 Normal mode
 Oscillation theory
 Selfadjoint
 Variation of parameters
 Spectral theory of ordinary differential equations
 Atkinson–Mingarelli theorem
References
 ↑ Pryce, J. D. (1993). Numerical Solution of Sturm–Liouville Problems. Oxford: Clarendon Press. ISBN 0198534159.
 ↑ Ledoux, V.; Van Daele, M.; Berghe, G. Vanden (2009). "Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics". Comput. Phys. Comm. 180: 532–554. doi:10.1016/j.cpc.2008.10.001.
 1 2 Kravchenko, V. V.; Porter, R. M. (2010). "Spectral parameter power series for Sturm–Liouville problems". Mathematical Methods in the Applied Sciences. 33 (4): 459–468. doi:10.1002/mma.1205.
Further reading
 Hazewinkel, Michiel, ed. (2001), "SturmLiouville theory", Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Hartman, Philip (2002). Ordinary Differential Equations (2 ed.). Philadelphia: SIAM. ISBN 9780898715101.
 Polyanin, A. D. & Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations (2 ed.). Boca Raton: Chapman & Hall/CRC Press. ISBN 1584882972.
 Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 9780821883280. (Chapter 5)
 Teschl, Gerald (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. Providence: American Mathematical Society. ISBN 9780821846605. (see Chapter 9 for singular S–L operators and connections with quantum mechanics)
 Zettl, Anton (2005). Sturm–Liouville Theory. Providence: American Mathematical Society. ISBN 0821839055.
 Birkhoff, Garrett (1973). A source book in classical analysis. Cambridge, Massachusetts: Harvard University Press. ISBN 0674822455. (See Chapter 8, part B, for excerpts from the works of Sturm and Liouville and commentary on them.)