APORTES AL CONOCIMIENTO

Many applications in physics and engineering involve the solution of lin- ear, second-order differential equations, which fall into a class of problems known as Sturm-Liouville eigenvalue problems. Such problems consist of a linear, second-order, ordinary differential equation containing a parame- ter whose value is determined so that the solution to the equation satisfies a given boundary condition. Thus, Sturm-Liouville problems are special kinds of boundary value problems for certain types of ordinary differential equations. The set of orthogonal functions generated by the solution to such problems may be used as basis functions for the Fourier expansion method of solving partial differential equations as well as other impor- tant problems. The special functions we briefly introduce in this chapter arise from one or another Sturm-Liouville problem and will be discussed in more detail later in the book.

To begin, we notice that a homogeneous linear second-order ordinary differential equation, containing the parameterλ multiplied by the function y(x),

a(x)y′′(x) + b(x)y′(x) + c (x)y(x) + λd(x)y(x) = 0, (2.1)

can be written in the Sturm-Liouville form d

dxp (x)y

80 2. Sturm-Liouville Theory where p (x) = e R b(x) a(x)dx_{, q (x) =} p (x)c (x) a(x) , r(x) = p (x)d(x) a(x) (2.3)

(clearly _{a(x) 6= 0). Here y(x) may represent some physical quantity in} which we are interested, such as the amplitude of a wave at a particu- lar location or the temperature at a particular time or location. The other functions in the equation express the physical situation that governs the behavior of the quantityy(x).

Reading Exercise. Show the equivalence of Equations (2.1) and (2.2): 1. Verify that substitution ofp (x) from Equations (2.3) into Equation

(2.2) gives Equation (2.1);

2. Verify that Equation (2.1) can be written in Sturm-Liouville form by dividing bya(x) and then multiplying by the integrating factor p (x) from Equation (2.3) .

As we will see, many physical problems result in the linear ordinary equation (2.2), where the functiony(x) is defined on an interval [a, b] and obeys homogeneous boundary conditions of the form

α1y′+β1y|x=a = 0,

α2y′+β2y|x=b = 0. (2.4)

Note that the constantsα1andβ1cannot both be zero simultaneously, nor

can the constantsα2 andβ2. Note also that the relative signs forαk and βk are not arbitrary; we generally must haveβ1/α1 < 0 and β2/α2 > 0.

This choice of signs (details of which are discussed in the Section 2.2.) is necessary in setting up boundary conditions for various classes of physi- cal problems. The very rare cases for which the signs are different occur in problems where there is explosive behavior, such as an exponential temper- ature increase. Throughout the book, we consider to be “normal” physical situations in which processes occur smoothly and thus the parameters in boundary conditions are restricted as above.

Equations (2.2) and (2.4) define a Sturm-Liouville problem. Solving this problem involves determining the values of the constantλ for which nontrivial solutionsy(x) exist. If αk = 0, the boundary condition simpli- fies toy = 0 (known as the Dirichlet boundary condition); if βk = 0, the

boundary condition isy′ _{= 0 (called the Neumann boundary condition);}

otherwise, the boundary condition is referred to as a mixed boundary con-

dition.

Notice that for the functiony(x) defined on an infinite or semi-infinite interval, the conditions of Equations (2.4) may not be specified and are of- ten replaced by the condition of regularity or physically reasonable behav- ior asx → ±∞. For instance, on the interval [a, ∞), the second condition in Equations (2.4) may not specified but can instead be replaced by the condition of regularity at_{x → ∞ (e.g., that y(∞) be finite).}

For the following discussion,p (x), q (x), r(x), and p′_{(x) are continu-}

ous, real functions on an interval [a, b] and p (x) ≥ 0 and r(x) ≥ 0 on the interval [a, b]. The coefficients αk andβk in Equations (2.4) are assumed to be real and independent ofλ.

The differential equation (2.2) and boundary conditions (2.4) are ho-

mogeneous, meaning that all the terms depend in some way ony(x), which is essential for the subsequent development. The trivial solutiony(x) = 0 is always possible for homogeneous equations, but we seek special val- ues ofλ (called eigenvalues) for which there are nontrivial solutions y(x) (called eigenfunctions) that depend onλ.

If we introduce the differential operator (called the Sturm-Liouville op-

erator)

Ly(x) = −_dxd p (x)y′(x) − q(x)y(x) = −p(x)y′′(_{x) − p}′(x)y′(_{x) − q (x)y(x),} (2.5) then Equation (2.2) becomes

Ly(x) = λr(x)y(x). (2.6)

As seen from Equation (2.5),Ly(x) is a linear operator. When r(x) = 1, this equation appears as an ordinary eigenvalue problem for which we must determineλ and y(x). For r(x) 6= 1, we have a similar problem, and the functionr(x) is called a weight function. As stated previously, the only requirement onr(x) is that it is real and nonnegative.

Equation (2.2) (or 2.6)) and boundary conditions (2.4) constitute the

boundary value problem. Its eigenvalues,λn, are real, and the eigenfunc- tions form an orthogonal set_{yn(x)}. To prove it, let us write Equation (2.6) for two eigenfunctions,yn(x) and ym(x), and take the complex con- jugate of the equation forym(x). Notice that in spite of the fact that p (x),

82 2. Sturm-Liouville Theory

q (x), and r(x) are real, λ and y(x) can be complex. We have Lyn(x) = λnr(x)yn(x)

and

Ly_m∗(x) = λ∗_mr(x)y∗_m(x). Multiplying the first of these equations byy∗

m(x) and the second by yn(x), we then integrate both froma to b and subtract the two results to obtain b Z a y_m∗(x)Lyn(x)dx − b Z a yn(x)Ly∗m(x)dx = λn− λ∗m
b Z a r(x)y_m∗(x)yn(x)dx. (2.7)

Using the definition ofL given by Equation (2.5), the left side of Equation (2.7) is  p (x) dy ∗ m dx · yn(x) − y ∗ m(x) · dyn dx  b a . (2.8)

Reading Exercise. Verify the previous statement.

Then, using the boundary conditions of Equations (2.4), we can easily prove that the expression in Equation (2.8) equals zero.

Reading Exercise. Verify that the expression in Equation (2.8) equals zero. Thus, we are left with

b Z a y_m∗(x)Lyn(x)dx = b Z a yn(x)Ly_m∗(x)dx. (2.9)

An operator, L, that satisfies Equation (2.9) may be termed a Hermitian or self-adjoint operator. Thus we may say that the Sturm-Liouville lin-

ear operator satisfying homogeneous boundary conditions is Hermitian.

Many important operators in physics, especially in quantum mechanics, are Hermitian. The main properties of Hermitian operators are that their eigenvalues are real and their eigenfunctions are orthogonal (or can be chosen to be orthogonal). We now supply the proof of these properties.

Using Equation (2.9), Equation (2.7) may be written as

λn_{− λ}∗_m
b Z a r(x)y_m∗(x)yn(x)dx = 0. (2.10)

When m = n, the integral cannot be zero (recall that r(x) > 0); thus, λ∗_n = _{λn and we have proved that the eigenvalues of a Sturm-Liouville} problem are real.

Then, forλm 6= λn, Equation (2.10) is b

Z

a

r(x)y∗_m(x)yn(x)dx = 0, (2.11)

and we conclude that the eigenfunctions corresponding to different eigen-

values of a Sturm-Liouville problem are orthogonal (with the weight func- tionr(x)).

The squared norm of the eigenfunctionyn(x) is defined to be

kynk2= b

Z

a

r(x)|yn(_x)|2dx. (2.12)

Note that the eigenfunctions of Hermitian operators always can be chosen

to be real. This can be done by creating linear combinations of the func-

tionsyn(_{x); for example, choose sin x and cos x instead of exp (±ix) for} the solutions of the equationy′′ +y = 0. Real eigenfunctions are more convenient to work with because it easier to match them to boundary con- ditions that are intrinsically real since they represent physical restrictions.

The above proof fails if λm = λn for some m 6= n (in other words there exist different eigenfunctions belonging to the same eigenvalues), in which case we cannot conclude that the corresponding eigenfunctions, ym(x) and yn(x), are orthogonal (although in some cases they are). If there are f eigenfunctions that have the same eigenvalues, we have an f-fold

degeneracy of the eigenvalue. In general, a degeneracy reflects a symmetry

of the underlying physical system (examples will be given later in this chapter). For a Hermitian operator it is always possible to construct linear combinations of the eigenfunctions belonging to the same eigenvalue so that these new functions are orthogonal.

Ifp (a) 6= 0 and p (b) 6= 0 then p (x) > 0 on the closed interval [a, b] (which follows from_{p (x) ≥ 0 for a ≤ x ≤ b) and we have the so-called}

regular Sturm-Liouville problem. Ifp (a) = 0, then we do not have the first of the boundary conditions in Equations (2.4); instead, we require y(x) andy′(x) to be finite at x = a. Similar situations occur if p (b) = 0, or if

84 2. Sturm-Liouville Theory

bothp (a) = 0 and p (b) = 0. All these cases correspond to the so-called

singular Sturm-Liouville problem.

Ifp (a) = p (b) and also instead of Equations (2.4) we have periodic

boundary conditions y(a) = y(b) and y′_{(a) = y}′_{(b), we have what is}

referred to as the periodic Sturm-Liouville problem.

The following list summarizes the three types of Sturm-Liouville prob- lems:

1. Forp (x) > 0 and r(x) > 0, we have the regular problem; 2. Forp (x) ≥ 0 and r(x) ≥ 0, we have the singular problem; 3. Forp (a) = p (b) and r(x) > 0, we have the periodic problem.

Notice that the interval (a, b) can be infinite, in which case the Sturm- Liouville problem is also classified as singular.

In this book, we discuss orthogonal polynomials, all of which are the orthogonal eigenfunctions of a corresponding Sturm-Liouville problem. Table 2.1 presents the Sturm-Liouville form of the differential equation defining these orthogonal functions and the Bessel functions.

In Table 2.1, the Bessel functions are denoted as Jn(x) and the Neu- mann functions asNn(x) (a commonly used alternative notation is Yn(x)). The interval for both can be infinite, [0, ∞) for Jn(x) and (0, ∞) for Nn(x), because the functionsNn(x) diverge as x → 0+. The Legendre polyno- mials are denoted byPn(x). The Chebyshev polynomials of the first kind are denoted asTn(x) and Chebyshev polynomials of the second kind are denoted asUn(x). The Jacobi polynomials are denoted as Pn(α ,β )(x), the Laguerre polynomials asL(_nα )(x) and the Hermite polynomials as Hn(x).

Theorem 2.1 lists several important properties of the Sturm-Liouville

problem. Theorem 2.1.

1. Each regular and each periodic Sturm-Liouville problem has an in- finite number of nonnegative, discrete eigenvalues 0 _{≤ λ}1 < λ2 < λ3 < ... < λn < ... such that λn → ∞ as n → ∞. All eigenvalues

are real numbers.

2. For each eigenvalue of a regular Sturm-Liouville problem, there is only one eigenfunction; for a periodic Sturm-Liouville problem, this property does not hold.

Function (n = 0, 1, . . .) Equation Weight function r(x) Range of x Jn(x) and Nn(x) _dxd xy′ +  λx − n2 x  y = 0 x [0, b] and (0, b] Pn(x) _dxd  1− x2
_y′_{ + λy = 0 1 [−1, 1]} Tn(x) _dxd hp 1_{− x}2_y′i+_λ√y 1−x2 = 0 1/ p 1_{− x}2 _{[−1, 1]} Un(x) _dxd h 1_{− x}2
3/2 y′i+_λp1_{− x}2_{y = 0} p_{1− x}2 _{[−1, 1]} Pn(α ,β )(x) (α , β > −1) d dx(1 − x) α +1_{(1 +x)}β +1_y′ +_{λ (1 − x)}α (1 +x)βy = 0 (1− x) α _{(1 +x)}β ₍ −1, 1) L_n(α )(_{x) (α > −1) dx}d xα +1e−x_y′_{ + λx}α_e−x_{y = 0 x}α_e−x _{[0, ∞)} Hn(x) _dxd h e−x2 y′i+_λe−x2 y = 0 e−x2 (_{−∞, ∞)}

Table 2.1.Sturm-Liouville equations.

3. For each of the types of Sturm-Liouville problems, the eigenfunctions corresponding to different eigenvalues are linearly independent. 4. For each of the types of Sturm-Liouville problems, the set of eigen-

functions is orthogonal with respect to the weight functionr(x) on

the interval [a, b].

5. If_{q (x) ≤ 0 on [a, b] and β}1/α1 < 0 and β2/α2> 0, then all λn ≥ 0.

Some of these properties have been proven previously, such as prop- erty (4) and part of property (1). The remaining part of property (1) should be obvious and will be shown in several examples, as well as property (5). Property (2) can be proved by postulating that there are two eigenfunctions corresponding to the same eigenvalue. We then apply Equations (2.2) and (2.4) to show that these two eigenfunctions coincide or differ at most by some multiplicative constant. We leave this proof to the reader as a reading exercise. Similarly, property (3) is easily proven.

Bessel functions and the orthogonal polynomials arise from singular Sturm-Liouville problems; thus, the first statement in the Theorem 2.1 is not directly applicable to these important cases. In spite of that, singular Sturm-Liouville problems may also have an infinite sequence of discrete eigenvalues, which we will later verify directly for Bessel functions and for the orthogonal polynomials. Notice that there is the possibility for a singu- lar Sturm-Liouville problem to have a continuous range of eigenvalues—in other words, a continuous spectrum—however, we will not encounter such situations in the problems we study in this book.

86 2. Sturm-Liouville Theory

When Equation (2.11) is satisfied, eigenfunctions yn(x) form a

complete orthogonal set on [a, b]. This means that any reasonable well- behaved function,f (x), defined on [a, b] can be expressed as a series of eigenfunctions (called a generalized Fourier series) of a Sturm-Liouville problem, in which case we may write

f (x) =

∞

X

n

anyn(x), (2.13)

where it is convenient to start the summation in some cases withn = 1 and in other cases withn = 0. An expression for the coefficients an can be found by multiplying both sides of Equation (2.13) byr(x)yn(x) and integrating over [a, b] to give

an = b R a r(x)f (x)yn(x)dx kynk2 . (2.14)

As we discussed in Chapter 1 on Fourier series, the sum in Equation (2.13) converges tof (x) in the mean, or

lim N→∞ b Z a r(x) " f (x) − N X n anyn(x) #2 dx. (2.15)

If we now substitutean from Equation (2.14) into Equation (2.13), we have f (x) = ∞ X n yn(x) kynk2 b Z a r(x′)f (x′)y_n(x′)dx′.

By interchanging the sum and the integral, we obtain

f (x) = b Z a f (x′)r(x′) ∞ X n yn(x)yn(x′) kynk2 dx′.

From here, we conclude that

∞ X n yn(x)yn(x′) kynk2 r(x′) =_{δ (x − x}′), (2.16)

whereδ (x) is the Dirac delta function, properties of which are discussed in Chapter 1 (see Equation (1.120)). Equation (2.16) is referred to as the completeness relation for the set of eigenfunctionsyn(x).

The Sturm-Liouville theory also provides a theorem for convergence of the series in Equation (2.13) at every pointx of [a, b]:

Theorem 2.2. Let_{yn(x)} be the set of eigenfunctions of a regular Sturm-

Liouville problem, and letf (x) and f′(x) be piecewise continuous on a

closed interval. Then the series Equation (2.13) converges tof (x) at every

point wheref (x) is continuous and to the value [f (x0+ 0) +f (x0− 0)]/2

ifx0is a point of discontinuity.

Theorem 2.2 is also valid for the orthogonal polynomials and Bessel functions related to singular Sturm-Liouville problems. This theorem, which is extremely important for applications, is similar to Theorem 1.3 for trigonometric Fourier series, which was discussed in Chapter 1.

Formulas in Equations (2.11) through (2.14) can be written in a more convenient way if we define a scalar product of real eigenfunctionsϕ and ψ as the number given by

ϕ · ψ = b

Z

a

r(x)ϕ(x)ψ (x)dx. (2.17)

This definition of the scalar product has properties identical to those for vectors in linear Euclidian space, a result that can be easily proved:

ϕ · ψ = ψ · ϕ,

(aϕ) · ψ = a(ϕ · ψ), (where a is a number) ϕ · (aψ) = aϕ · ψ,

ϕ · (ψ + φ) = ϕ · ψ + ϕ · φ, ϕ · ϕ ≥ 0.

(2.18)

The last property relies on the assumption made for the Sturm-Liouville equation that_{r(x) ≥ 0. If ϕ is continuous on [a, b], then ϕ · ϕ = 0 only if} ϕ is zero.

Reading Exercise. Prove the relations given in Equations (2.18).

88 2. Sturm-Liouville Theory

In terms of the scalar product, the orthogonality of eigenfunctions (de- fined by Equation (2.11)) means that

yn· ym = 0 if n 6= m (2.19) and the formula for the Fourier coefficients in Equation (2.14) becomes

an = f · y n yn· yn