There is a physically very important class of operators with a weight function. These occur in the socalled SturmLiouville equations, which are eigenvalue equations of the form
where ρ(x) > 0 is a given real positive weight function and the operator L is of the special SturmLiouville type,
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
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 SL 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.
Given a general second order differential equation, that we suspect might be written as SturmLiouville 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 SturmLiouville equation. Suppose first that we are given the function p(x) in the SturmLiouville 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),
We have thus found the function p to bring it to SturmLiouville form. The function ρ = τp must be positive, and thus since p is positive, τ must be positive.
Name  p(x)  q(x)  ρ(x)  [a,b] 
Legendre’s equation  (1 − {x}^{2})  0  1  [−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}∕x  x  [0,R], R finite. 
and many others.  
There are many wellknown examples in physics, see Table 3.1. Almost all cases we meet in physics are Hermitian SturmLiouville operators. Some of these will be investigated further below, but first we need a useful property of SturmLiouville operators
In general, one can show that for any two real functions u(x), v(x) defined for x ∈ [a,b], and a SturmLiouville operator L also defined on [a,b],
After integration we thus conclude that
From the useful identity (3.9) we can draw some interesting conclusions about Hermitian SturmLiouville operators. By definition, an operator L is Hermitian if
for any two vectors u, v in the space. Hence, from this and (3.9), a SL operator is Hermitian if and only if the boundary conditions at a and b are such that
where the Wronskian W is defined as
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.
Since a SturmLiouville 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
We now multiply (3.10) by v(x) and (3.11) by u(x), and subtract:
}
Hence
i.e.,
Since u is known, this is differential equation for v (first order!). The technique applicable is the integrating factor or substitution of v = uw,
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 SturmLiouville operator is defined, is delimited by such special singular points, and p(a) = p(b) = 0!
Consider a second order differential equation
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.
For Hermitian SL operators, we state witout proof that:
Since the SL 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.
is unbounded, {λ}_{n} →∞ as n →∞ .
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.)