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
where is a given real positive weight function and the operator is of the special Sturm-Liouville type,
where , are given real functions and 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,
(3.6) |
Many equations can be put in S-L form by multiplying by a suitably chosen function , 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 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,
so let us assume an equation of that form.
We shall show below that for a Sturm-Liouville equation. Suppose first that we are given the function in the Sturm-Liouville operator. We can then multiply both sides of the equation with , and find
If we compare this with equation ( 3.6 ) above we see that
(3.7) |
If we do not know , we can solve ( 3.7 ) for ,
We have thus found the function to bring it to Sturm-Liouville form. The function must be positive, and thus since is positive, must be positive.
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Legendre’s equation | ||||
Laguerre’s equation | ||||
Hermite’s equation | ||||
Chebychev’s equations | ||||
Bessel’s equation | , finite. | |||
and many others. | ||||
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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
In general, one can show that for any two real functions , defined for , and a Sturm-Liouville operator also defined on ,
After integration we thus conclude that
From the useful identity ( 3.9 ) we can draw some interesting conclusions about Hermitian Sturm-Liouville operators. By definition, an operator is Hermitian if
for any two vectors , in the space. Hence, from this and ( 3.9 ), a S-L operator is Hermitian if and only if the boundary conditions at and are such that
where the Wronskian is defined as
In mathematical physics the domain is often delimited by points and where . If we then add a boundary condition that and are finite (or a specific finite number) as for all solutions , the operator is Hermitian.
Note that such boundary conditions forbid “second solutions” in general – see next section.
Since a Sturm-Liouville equation is by definition second order, there are two independent solutions. If we have already obtained one (finite) solution for a given , we would like to know the second solution, which we call . Thus
We now multiply ( 3.10 ) by and ( 3.11 ) by , and subtract:
Hence
i.e.,
Since is known, this is differential equation for (first order!). The technique applicable is the integrating factor or substitution of ,
We can of course add a constant to , but that would just add a component proportionsl to into the solution, which we already know is allowed. We can also take , since the multiplication with is a trivial reflection of linearity.
These solutions do not exist (i.e., diverge) for points such that , which are called singular points. This may sound like a superficial remark, but almost always the interval , on which the Sturm-Liouville operator is defined, is delimited by such special singular points, and !
Consider a second order differential equation
If at a point or diverges, but and are finite, is called a regular singular point. If diverges faster than and/or diverges faster than we speak of an irregular singular point.
For Hermitian S-L operators, we state witout proof that:
Since the S-L equation is real and its solution for any eigenvalue is unique, this implies apart from a multiplicative constant. Hence one can (and we will) always choose real eigenfunctions.
is unbounded, as .
form a complete basis set of functions on the interval satisfying the boundary conditions. (Proof given in the Variational Calculus section, but not necessarily discussed in class.)