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3.3 Sturm-Liouville equations

There is a physically very important class of operators with a weight function. These occur in the so-called Sturm-Liouville equations, which are eigenvalue equations of the form

Ly(x) = λρ(x)y(x),

where ρ(x) > 0 is a given real positive weight function and the operator L is of the special Sturm-Liouville type,

L = − {d\over dx}\left (p(x) {d\over dx}⋅\right ) + q(x)

where p(x), q(x) are given real functions and p(x) is positive. The dot denotes the place the argument of the operator must be inserted. Explicitly, using (3.3) and (3.3), we see that they are homogeneous second order equations of the form

− {d\over dx}\left (p(x) {d\over dx}y(x)\right ) + q(x)y(x) − λρ(x)y(x) = 0,

or equivalently, expanding out the derivatives,

p(x){{d}^{2}y\over d{x}^{2}} + {dp\over dx} {dy\over dx} − q(x)y(x) + λρ(x)y(x) = 0.
(3.6)

Many equations can be put in S-L form by multiplying by a suitably chosen function α(x), which is determined by requiring a differential equation of the form Eq. (3.6), see the next section.

3.3.1 How to bring an equation to Sturm-Liouville form

Given a general second order differential equation, that we suspect might be written as Sturm-Liouville equation, how do we find out whether this is true?

We start from a “canonical form”. It is straightforward to rewrite any second order differential equation so that the coefficient of the second derivative is 1,

y''(x) + α(x)y'(x) + β(x)y(x) + λτ(x)y(x) = 0,

so let us assume an equation of that form.

We shall show below that τ > 0 for a Sturm-Liouville equation. Suppose first that we are given the function p(x) in the Sturm-Liouville operator. We can then multiply both sides of the equation with p, and find

p(x)y''(x) + p(x)α(x)y'(x) + p(x)β(x)y(x) + λp(x)τ(x)y(x) = 0.

If we compare this with equation (3.6) above we see that

p'(x) = α(x)p(x),\qquad q(x) = −β(x)p(x),\qquad ρ(x) = τ(x)p(x).
(3.7)

If we do not know p, we can solve (3.7) for p(x),

\eqalignno{ {d\mathop{ln}\nolimits (p(x))\over dx} & = α(x), & & \cr p(x) & =\mathop{ exp}\nolimits \left ({\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }^{x}α(x')dx'\right ). & & }

We have thus found the function p to bring it to Sturm-Liouville form. The function ρ = τp must be positive, and thus since p is positive, τ must be positive.

Table 3.1: A few well-known examples of Sturm-Liouville problems that occur in mathematical physics

Name p(x)q(x)ρ(x)[a,b]
Legendre’s equation (1 − {x}^{2})01[−1,1]
Laguerre’s equation x{e}^{−x}0{e}^{−x}[0,∞)
Hermite’s equation {e}^{−{x}^{2} }0{e}^{−{x}^{2} }(−∞,∞)
Chebychev’s equations{(1 − {x}^{2})}^{1∕2}0{(1 − {x}^{2})}^{−1∕2}[−1,1]
Bessel’s equation x− {ν}^{2}∕xx[0,R], R finite.
and many others.

There are many well-known examples in physics, see Table 3.1. Almost all cases we meet in physics are Hermitian Sturm-Liouville operators. Some of these will be investigated further below, but first we need a useful property of Sturm-Liouville operators

3.3.2 A useful result

In general, one can show that for any two real functions u(x), v(x) defined for x ∈ [a,b], and a Sturm-Liouville operator L also defined on [a,b],

\eqalignno{ vLu − (Lv)u & = −v(x) {d\over dx}\left (p(x) {d\over dx}u(x)\right ) + v(x)q(x)u(x) & & \cr \qquad &\quad + \left [ {d\over dx}\left (p(x) {d\over dx}v(x)\right )\right ]u(x) − v(x)q(x)u(x) & & \cr & = −v(pu')' + u(pv')' & & \cr & ={\bigl [ −vpu' + upv'\bigr ]}'. &\text{(3.8)} }

After integration we thus conclude that

\eqalignno{ {\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{a}^{b}dx\kern 1.66702pt [vLu − (Lv)u] & = (v,Lu) − (u,Lv) & & \cr & ={\bigl [ p{(uv' − vu')\bigr ]}}_{a}^{b}. &\text{(3.9)} }

3.3.3 Hermitian Sturm Liouville operators

From the useful identity (3.9) we can draw some interesting conclusions about Hermitian Sturm-Liouville operators. By definition, an operator L is Hermitian if

{\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{a}^{b}dx\kern 1.66702pt [vLu − (Lv)u] = 0

for any two vectors u, v in the space. Hence, from this and (3.9), a S-L operator is Hermitian if and only if the boundary conditions at a and b are such that

{\bigl [p{(uv' − vu')\bigr ]}}_{a}^{b} = p(b)W(b) − p(a)W(a),

where the Wronskian W is defined as

W(x) = \left (\array{ u(x)&v(x)\cr u'(x) &v'(x) } \right ) = u'(x)v(x)−u(x)v'(x).

In mathematical physics the domain is often delimited by points a and b where p(a) = p(b) = 0. If we then add a boundary condition that w(x)p(x) and w'(x)p(x) are finite (or a specific finite number) as x → a,b for all solutions w(x), the operator is Hermitian.

Note that such boundary conditions forbid “second solutions” in general – see next section.

3.3.4 Second solutions, singularities

Since a Sturm-Liouville equation is by definition second order, there are two independent solutions. If we have already obtained one (finite) solution u(x) for a given λ, we would like to know the second solution, which we call v(x). Thus

\begin{eqnarray} Lu(x) + λρu(x)& =& 0,%&(3.10) \\ Lv(x) + λρv(x)& =& 0.%&(3.11) \\ \end{eqnarray}

We now multiply (3.10) by v(x) and (3.11) by u(x), and subtract:

\eqalignno{ uLv − vLu & = 0 & & \text{or, using the result above} \cr {d\over dx}{\bigl [puv' − pvu'\bigr ]} = 0. & &

}

Hence

puv' − pvu' = c\quad ,

i.e.,

uv' − vu' = {c\over p(x)}\quad .

Since u is known, this is differential equation for v (first order!). The technique applicable is the integrating factor or substitution of v = uw,

\eqalignno{ uu'w + uuw' − uu'w & = c∕p\Rightarrow & & \cr w' & = {c\over p{u}^{2}}\Rightarrow & & \cr w(x) & = c{\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }^{x} {1\over p(x')u{(x')}^{2}}dx'\quad . & & }

We can of course add a constant to w, but that would just add a component proportionsl to u into the solution, which we already know is allowed. We can also take c = 1, since the multiplication with c is a trivial reflection of linearity.

These solutions do not exist (i.e., diverge) for points such that p(x) = 0, which are called singular points. This may sound like a superficial remark, but almost always the interval [a,b], on which the Sturm-Liouville operator is defined, is delimited by such special singular points, and p(a) = p(b) = 0!

Consider a second order differential equation

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

If at a point x = {x}_{0} P(x) or Q(x) diverges, but (x − {x}_{0})P(x) and {(x − {x}_{0})}^{2}Q(x) are finite, {x}_{0} is called a regular singular point. If P(x) diverges faster than 1∕(x − {x}_{0}) and/or Q(x) diverges faster than 1∕{(x − {x}_{0})}^{2} we speak of an irregular singular point.

3.3.5 Eigenvectors and eigenvalues

For Hermitian S-L operators, we state witout proof that:

  1. The eigenvalues are real and non-degenerate, i.e., there exists only one finite solution {u}_{n}(x) for each eigenvalue {λ}_{n}.

    Since the S-L equation is real and its solution {u}_{n}(x) for any eigenvalue is unique, this implies {u}_{n}(x) = {u}_{n}^{∗}(x) apart from a multiplicative constant. Hence one can (and we will) always choose real eigenfunctions.

  2. There exists a lowest eigenvalue {λ}_{0} (this relies on the positivity of p(x)) and the sequence
    {λ}_{0} < {λ}_{1} < \mathop{\mathop{…}} < {λ}_{n} < \mathop{\mathop{…}}\quad .

    is unbounded, {λ}_{n} →∞ as n →∞ .

  3. The number of nodes in the n-th eigenvector, if the corresponding eigenvalues are ordered as above, is exactly equal to n.
  4. Eigenfunctions u, v with u\mathrel{≠}v are orthogonal with weight function ρ(x),
    {(u,v)}_{ρ} ={\mathop{ \mathop{\mathop{∫ }\nolimits }}\nolimits }_{a}^{b}dx\kern 1.66702pt ρ(x){u}^{∗}(x)v(x) = 0\quad .

  5. The eigenfunctions
    {u}_{1}(x),{u}_{2}(x),\mathop{\mathop{…}},{u}_{n}(x),\mathop{\mathop{…}}

    form a complete basis set of functions on the interval [a,b] satisfying the boundary conditions. (Proof given in the Variational Calculus section, but not necessarily discussed in class.)

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