Sturm-Liouville problems

The basic Sturm-Liouville problem is
-
∂
∂x
(p
′
y
)+qy=λy
with boundary conditions
α
1
y(a)+
α
2
′
y
(a)
=
0
β
1
y(b)+
β
2
′
y
(b)
=
0
.
If
p
,
′
p
, and
q
are continuous on
[a,b]
and
p
is never zero in
[a,b]
, then the problem is called regular. Otherwise, it is singular. A regular Sturm-Liouville problem is guaranteed to have a complete, orthonormal set of eigenfunctions, while a singular problem might not.
In the most basic example,
p(x)=1
and
q(x)=
α
2
=
β
2
=0
, so the eigenfunctions are linear combinations of trigonometric functions; this yields Fourier's theory. The purpose of this notebook is to illustrate the existence of other examples.

Assumptions


Example 1


Example II


Example III - Airy functions


Example IV - Bessel functions and weighted inner products
