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 second-order linear differential equation of the form

${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}\left[p(x){\frac {{\mathrm {d}}y}{{\mathrm {d}}x}}\right]+q(x)y=-\lambda w(x)y,$

(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 non-trivial 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 each such 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

$\alpha _{1}y(a)+\alpha _{2}y'(a)=0\qquad \qquad \qquad (\alpha _{1}^{2}+\alpha _{2}^{2}>0),$

(2)

$\beta _{1}y(b)+\beta _{2}y'(b)=0\qquad \qquad \qquad (\beta _{1}^{2}+\beta _{2}^{2}>0),$

(3)

Under the assumption that the S–L problem is regular, the main tenet of Sturm–Liouville theory states that:

The eigenvalues λ₁, λ₂, λ₃, ... of the regular Sturm–Liouville problem (1)–(2)–(3) are real and can be ordered such that

\lambda _{1}<\lambda _{2}<\lambda _{3}<\cdots <\lambda _{n}<\cdots \to \infty ;

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 n-th fundamental solution satisfying the regular Sturm–Liouville problem (1)–(2)–(3).

The normalized eigenfunctions form an orthonormal basis

\int _{a}^{b}y_{n}(x)y_{m}(x)w(x)\,{\mathrm {d}}x=\delta _{{mn}},

in the Hilbert space L²([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 self-adjoint form. All second-order linear ordinary differential equations can be recast in the form on the left-hand side of (1) by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector.)

Examples

The Bessel equation

x^{2}y''+xy'+(x^{2}-\nu ^{2})y=0

which can be written in Sturm–Liouville form as

(xy')'+\left(x-{\frac {\nu ^{2}}{x}}\right)y=0.

The Legendre equation

(1-x^{2})y''-2xy'+\nu (\nu +1)y=0

which can easily be put into Sturm–Liouville form, since ${\tfrac {d}{dx}}(1-x^{2})=-2x$ , so the Legendre equation is equivalent to

\left((1-x^{2})y'\right)'+\nu (\nu +1)y=0

An example using an integrating factor

x^{3}y''-xy'+2y=0.

Divide throughout by x³:

y''-{\frac {1}{x^{2}}}y'+{\frac {2}{x^{3}}}y=0

Multiplying throughout by an integrating factor of

\mu (x)=e^{\int -{\frac {1}{x^{2}}}\mathrm {d} x}=e^{\frac {1}{x}},

gives

e^{{{\frac {1}{x}}}}y''-{\frac {e^{{{\frac {1}{x}}}}}{x^{2}}}y'+{\frac {2e^{{{\frac {1}{x}}}}}{x^{3}}}y=0

which can be easily put into Sturm–Liouville form since

{\frac {d}{dx}}e^{\frac {1}{x}}=-{\frac {e^{\frac {1}{x}}}{x^{2}}}

so the differential equation is equivalent to

\left(e^{\frac {1}{x}}y'\right)'+{\frac {2e^{\frac {1}{x}}}{x^{3}}}y=0.

The integrating factor for a general second-order differential equation

P(x)y''+Q(x)y'+R(x)y=0

multiplying through by the integrating factor

\mu (x)={\frac {1}{P(x)}}e^{\int {\frac {Q(x)}{P(x)}}\mathrm {d} x},

and then collecting gives the Sturm–Liouville form:

{\frac {d}{dx}}(\mu (x)P(x)y')+\mu (x)R(x)y=0

or, explicitly,

{\frac {d}{dx}}\left(e^{{\int {\frac {Q(x)}{P(x)}}{\mathrm {d}}x}}y'\right)+{\frac {R(x)}{P(x)}}e^{{\int {\frac {Q(x)}{P(x)}}\,{\mathrm {d}}x}}y=0

Sturm–Liouville equations as self-adjoint differential operators

The map

Lu=-{\frac {1}{w(x)}}\left({\frac {{\mathrm {d}}}{{\mathrm {d}}x}}\left[p(x){\frac {{\mathrm {d}}u}{{\mathrm {d}}x}}\right]+q(x)u\right)

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

Lu=\lambda u.

This is precisely the eigenvalue problem; that is, one is trying to find the eigenvalues λ₁, λ₂, λ₃, ... and the corresponding eigenvectors u₁, u₂, u₃, ... of the L operator. The proper setting for this problem is the Hilbert space L²([a, b], w(x) dx) with scalar product

\langle f,g\rangle =\int _{a}^{b}{\overline {f(x)}}g(x)w(x)\,\mathrm {d} x.

In this space L is defined on sufficiently smooth functions which satisfy the above boundary conditions. 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

(L-z)^{{-1}},\qquad z\in {\mathbb {C}},

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

(L-z)^{-1}u=\alpha u,\qquad Lu=(z+\alpha ^{-1})u,

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 one-dimensional time-independent 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:

$Lu=-{\frac {{\mathrm {d}}^{2}u}{{\mathrm {d}}x^{2}}}=\lambda u$

(4)

where the unknowns are λ and u(x). As above, we must add boundary conditions, we take for example

u(0)=u(\pi )=0.

Observe that if k is any integer, then the function

u(x)=\sin kx

is a solution with eigenvalue λ = k². 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

Lu=x,\qquad x\in (0,\pi )

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 ^ikxx dx or by consulting a table of Fourier transforms, that we thus obtain

Lu=\sum _{k=1}^{\infty }-2{\frac {(-1)^{k}}{k}}\sin kx.

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 L² 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

u=\sum _{{k=1}}^{{\infty }}2{\frac {(-1)^{k}}{k^{3}}}\sin kx.

In this case, we could have found the answer using anti-differentiation. This technique yields

u={\tfrac {1}{6}}(x^{3}-\pi ^{2}x),

whose Fourier series agrees with the solution we found. The anti-differentiation 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 ≤ L₁, 0 ≤ y ≤ L₂. The equation of motion for the vertical membrane's displacement, W(x, y, t) is given by the wave equation:

{\frac {\partial ^{2}W}{\partial x^{2}}}+{\frac {\partial ^{2}W}{\partial y^{2}}}={\frac {1}{c^{2}}}{\frac {\partial ^{2}W}{\partial t^{2}}}.

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/c²)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, L₁ or y = 0, L₂ and define the simplest possible S–L eigenvalue problems as in the example, yielding the "normal mode solutions" for W with harmonic time dependence,

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)

where m and n are non-zero integers, A_mn are arbitrary constants, and

\omega _{{mn}}^{2}=c^{2}\left({\frac {m^{2}\pi ^{2}}{L_{1}^{2}}}+{\frac {n^{2}\pi ^{2}}{L_{2}^{2}}}\right).

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 $\omega _{{mn}}$ . 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.

2. Finite difference method.

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

y(x)\int _{a}^{x}{\frac {{\mathrm {d}}t}{p(t)y(t)^{2}}}

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 λ₀^* (often λ₀^* = 0; it does not need to be an eigenvalue) and any solution y₀ of (1) with λ = λ₀^* which does not vanish on [a, b]. (Discussion below of ways to find appropriate y₀ and λ₀^*.) 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/(py₀²) and wy₀² and integrated, specifically

$X^{(n)}(t)={\begin{cases}\displaystyle -\int _{a}^{x}X^{(n-1)}(t)p(t)^{-1}y_{0}(t)^{-2}\,\mathrm {d} t&n{\text{ odd}},\\[6pt]\displaystyle \quad \int _{a}^{x}X^{(n-1)}(t)y_{0}(t)^{2}w(t)\,\mathrm {d} t&n{\text{ even}}\end{cases}}$

(5)

${\widetilde {X}}^{(n)}(t)={\begin{cases}\displaystyle \quad \int _{a}^{x}{\widetilde {X}}^{(n-1)}(t)y_{0}(t)^{2}w(t)\,\mathrm {d} t&n{\text{ odd}},\\[6pt]\displaystyle -\int _{a}^{x}{\widetilde {X}}^{(n-1)}(t)p(t)^{-1}y_{0}(t)^{-2}\,\mathrm {d} t&n{\text{ even}}\end{cases}}$

(6)

when n > 0. The resulting iterated integrals are now applied as coefficients in the following two power series in λ:

u_{0}=y_{0}\sum _{{k=0}}^{\infty }\left(\lambda -\lambda _{0}^{*}\right)^{k}\widetilde X^{{(2k)}},

u_{1}=y_{0}\sum _{{k=0}}^{\infty }\left(\lambda -\lambda _{0}^{*}\right)^{k}X^{{(2k+1)}}.

Then for any λ (real or complex), u₀ and u₁ 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 y₀.)

Next one chooses coefficients c₀, c₁ so that the combination y = c₀u₀ + c₁u₁ 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 u₀(b) and u₁(b) and the derivatives u₀'(b) and u₁'(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 sought-after eigenvalues.

When λ = λ₀, 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 y₀. 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 μ₀ = 0. While there is no guarantee that u₀ or u₁ will not vanish, the complex function y₀ = u₀ + iu₁ 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 y₀ of (1) for the value λ₀ = 0. In practice if (1) has real coefficients, the solutions based on y₀ 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:

f(x){\frac {\partial ^{2}u}{\partial x^{2}}}+g(x){\frac {\partial u}{\partial x}}+h(x)u={\frac {\partial u}{\partial t}}+k(t)u

u(a,t)=u(b,t)=0

u(x,0)=s(x)

Let us apply separation of variables, which in doing we must impose that:

u(x,t)=X(x)T(t)

Then our above PDE may be written as:

{\frac {{\hat {L}}X(x)}{X(x)}}={\frac {{\hat {M}}T(t)}{T(t)}}

where

{\hat {L}}=f(x){\frac {{\mathrm {d}}^{2}}{{\mathrm {d}}x^{2}}}+g(x){\frac {{\mathrm {d}}}{{\mathrm {d}}x}}+h(x),\qquad {\hat {M}}={\frac {{\mathrm {d}}}{{\mathrm {d}}t}}+k(t)

Since, by definition, ${\hat {L}}$ and $X(x)$ are independent of time t and ${\hat {M}}$ and $T(t)$ are independent of position x, then both sides of the above equation must be equal to a constant:

{\hat {L}}X(x)=\lambda X(x)

X(a)=X(b)=0\,

{\hat {M}}T(t)=\lambda T(t)\,

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 $X_{n}(x)$ and eigenvalues $\lambda _{n}$ . The second of these equations can be analytically solved once the eigenvalues are known.

{\frac {{\mathrm {d}}}{{\mathrm {d}}t}}T_{n}(t)=(\lambda _{n}-k(t))T_{n}(t)

T_{n}(t)=a_{n}e^{{-\left(\lambda _{n}t-\int _{0}^{t}k(\tau ){\mathrm {d}}\tau \right)}}

u(x,t)=\sum _{n}a_{n}X_{n}(x)e^{{-\left(\lambda _{n}t-\int _{0}^{t}k(\tau ){\mathrm {d}}\tau \right)}}

a_{n}={\frac {\langle X_{n}(x),s(x)\rangle }{\langle X_{n}(x),X_{n}(x)\rangle }}