Sturm–Liouville theory


In mathematics and its applications, classical Sturm–Liouville theory is the theory of real second-order linear ordinary differential equations of the form:
for given coefficient functions ,, and and an unknown function of the free variable. The function, sometimes denoted, is called the weight or density function. All second-order linear ordinary differential equations can be reduced to this form.
In the simplest case where all coefficients are continuous on the finite closed interval and has continuous derivative, a function is called a solution if it is continuously differentiable on and satisfies the equation ' at every point in. In addition, is typically required to satisfy some boundary conditions at and. Each such equation ' together with its boundary conditions constitutes a Sturm-Liouville problem.
The value of is not specified in the equation: finding the for which there exists a non-trivial solution is part of the given S-L problem. Such values of, when they exist, are called the eigenvalues of the problem, and the corresponding solutions are the eigenfunctions associated to each. This terminology is because the solutions correspond to the eigenvalues and eigenfunctions of a Hermitian differential operator in an appropriate function space. Sturm–Liouville theory studies the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in the function space.
This theory is important in applied mathematics, where S-L problems occur very commonly, particularly when dealing with separable linear partial differential equations. For example, in quantum mechanics, the one-dimensional time-independent Schrödinger equation is a S-L problem.
A Sturm-Liouville problem is said to be regular if,, and,,, are continuous functions over the finite interval, and the problem has separated boundary conditions of the form:
The main result of Sturm–Liouville theory states that, for the regular Sturm–Liouville problem ',',:
The theory is named after Jacques Charles François Sturm and Joseph Liouville.

Reduction to Sturm–Liouville form

The differential equation ' is said to be in Sturm–Liouville form or self-adjoint form. All second-order linear ordinary differential equations can be recast in the form on the left-hand side of ' by multiplying both sides of the equation by an appropriate integrating factor. Some examples are below.

[Bessel equation]

which can be written in Sturm–Liouville form as

Legendre equation">Legendre polynomials">Legendre equation

which can easily be put into Sturm–Liouville form, since, so the Legendre equation is equivalent to

Example using an integrating factor

Divide throughout by :
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

Integrating factor for general second-order equation

Multiplying through by the integrating factor
and then collecting gives the Sturm–Liouville form:
or, explicitly:

Sturm–Liouville equations as self-adjoint differential operators

The mapping defined by:
can be viewed as a linear operator mapping a function to another function, and it can be studied in the context of functional analysis. In fact, equation can be written as
This is precisely the eigenvalue problem; that is, one seeks eigenvalues and the corresponding eigenvectors of the operator. The proper setting for this problem is the Hilbert space Lp space#Weighted Lp spaces| with scalar product
In this space is defined on sufficiently smooth functions which satisfy the above regular boundary conditions. Moreover, L is 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 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 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. As a consequence of the Arzelà–Ascoli theorem, this integral operator is compact and existence of a sequence of eigenvalues which converge to 0 and eigenfunctions which form an orthonormal basis follows from the spectral theorem for compact operators. Finally, note that
are equivalent, so we may take with the same eigenfunctions.
If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls 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. This is important in quantum mechanics, since the one-dimensional time-independent Schrödinger equation is a special case of a S-L equation.

Application to inhomogeneous second-order boundary value problems

Consider a general inhomogeneous second-order linear differential equation
for given functions. As before, this can be reduced to the S-L form : writing a general S-L operator as:
one solves the system:
It suffices to solve the first two equations, which amounts to solving, or
A solution is:
Given this transformation, one is left to solve:
In general, if initial conditions at some point are specified, for example and, a second order differential equation can be solved using ordinary methods and the Picard–Lindelöf theorem ensures that the differential equation has a unique solution in a neighbourhood of the point where the initial conditions have been specified.
But if in place of specifying initial values at a single point, it is desired to specify values at two different points, e.g. and, the problem turns out to be much more difficult. Notice that by adding a suitable known differentiable function to, whose values at and satisfy the desired boundary conditions, and injecting inside the proposed differential equation, it can be assumed without loss of generality that the boundary conditions are of the form and.
Here, the Sturm–Liouville theory comes in play: indeed, a large class of functions can be expanded in terms of a series of orthonormal eigenfunctions of the associated Liouville operator with corresponding eigenvalues :
Then a solution to the proposed equation is evidently:
This solution will be valid only over the open interval, and may fail at the boundaries.

Example: Fourier series

Consider the Sturm–Liouville problem:
for the unknowns are and. For boundary conditions, we take for example:
Observe that if is any integer, then the function
is a solution with eigenvalue. 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 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 expand as a Fourier series. The reader may check, either by integrating 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 "square-summable", the Fourier series converges in 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 series converge at every point of differentiability, and at jump points converges to the average of the left and right limits.
Therefore, by using formula , we obtain the solution:
In this case, we could have found the answer using antidifferentiation, but this is no longer useful in most cases when the differential equation is in many variables.

Application to [partial differential equation]s

Normal modes

Certain partial differential equations can be solved with the help of S-L theory. Suppose we are interested in the vibrational modes of a thin membrane, held in a rectangular frame,,. The equation of motion for the vertical membrane's displacement, is given by the wave equation:
The method of separation of variables suggests looking first for solutions of the simple form. For such a function the partial differential equation becomes. Since the three terms of this equation are functions of separately, they must be constants. For example, the first term gives for a constant . The boundary conditions are when, or, and define the simplest possible S-L eigenvalue problems as in the example, yielding the "normal mode solutions" for with harmonic time dependence,
where and are non-zero integers, are arbitrary constants, and
The functions form a basis for the Hilbert space of solutions of the wave equation; that is, an arbitrary solution can be decomposed into a sum of these modes, which vibrate at their individual frequencies. This representation may require a convergent infinite sum.

Second-order linear equation

For a linear second-order in one spatial dimension and first-order in time of the form:
Separating variables, we assume that
Then our above partial differential equation may be written as:
where
Since, by definition, and are independent of time and and are independent of position, 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 in terms of the eigenfunctions and eigenvalues. The second of these equations can be analytically solved once the eigenvalues are known.
where

Representation of solutions and numerical calculation

The Sturm–Liouville differential equation with boundary conditions may be solved analytically, which can be exact or provide an approximation, by the Rayleigh–Ritz method, or by the matrix-variational method of Gerck et al.
Numerically, a variety of methods are also available. 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. These methods proceed by guessing a value of, solving an initial value problem defined by the boundary conditions at one endpoint, say,, of the interval, comparing the value this solution takes at the other endpoint with the other desired boundary condition, and finally increasing or decreasing as necessary to correct the original value. This strategy is not applicable for locating complex eigenvalues.
  2. Finite difference method.
  3. The spectral parameter power series method makes use of a generalization of the following fact about second-order ordinary differential equations: if is a solution that does not vanish at any point of, then the function

    Construction of a nonvanishing solution

The SPPS method can, itself, be used to find a starting solution. Consider the equation ; i.e.,,, and are replaced in ' by 0,, and respectively. Then the constant function 1 is a nonvanishing solution corresponding to the eigenvalue. While there is no guarantee that or will not vanish, the complex function 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 of ' for the value. In practice if has real coefficients, the solutions based on will have very small imaginary parts which must be discarded.