Chapter 10 Sturm-Liouville theory–
Orthogonal functions
December 5 Self-adjoint ODEs
Introduction:
We are going to study a standard second order ODE called the Sturm-Liouville ODE. The
solution of this ODE form a function space, or Hilbert space, which is similar to the vector
space we have known. We will use similar concepts as in vector spaces, such as inner
products, orthogonality, eigenvalues, eigenfunctions, and completeness. Sturm-Liouville
ODE has been used to generate various special functions in physics.
1
Self-adjoint operators:
The most general form of a linear second-order differential operator L is
d2
d
L  p0 ( x) 2  p1 ( x)  p2 ( x)
dx
dx
Lu ( x)  p0 ( x)u" ( x)  p1 ( x)u ' ( x)  p2 ( x)u ( x)
Here to apply this operator to our former results of differential equations from the
separation of variables, we assume p0, p1, and p2 be real functions. Also p0 > 0 in the range
of interest a< x <b, while p0=0 can happen at the boundaries.
We define the adjoint operator L as
d2
d
L  2 p0 ( x ) 
p1 ( x)  p2 ( x)
dx
dx
L u ( x)  ( p0u )"( p1u )' p2u  p0u"(2 p0 ' p1 )u '( p0 " p1 ' p2 )u
Now if p0 '  p1, we have
d 
du ( x) 
p
(
x
)
 q ( x)u ( x)
dx 
dx 
by changing p0 ( x)  p ( x), p2 ( x)  q( x).
L u ( x)  Lu ( x)  ( p0u ' )' p2u 
LL 
d 
d
p
(
x
)
 q( x)


dx 
dx 
2
Theorem: A non-self-adjoint linear second-order differential operator can always be
transformed into the self-adjoint form by Lw(x)L.
Therefore the theory on linear second-order self-adjoint differential equations, which
we are going to explore, is general.
Proof :
Lu ( x)  p0 ( x)u" ( x)  p1 ( x)u ' ( x)  p2 ( x)u ( x)
 x p

1
exp   1 dx L  w( x)L ( w( x)  0 is called the weight function. )
p0
 p0 
 x p

1
w( x)Lu ( x) 
exp   1 dx Lu ( x)
p0
 p0 
Transform L 
 x p

 x p

 x p

p
p
 exp   1 dx u" 1 exp   1 dx u ' 2 exp   1 dx u
p0
p0
 p0 
 p0 
 p0 
 x p1 
d   x p1  du  p2



dx   
exp  
dx u
exp  
dx   p0  dx  p0
 p0 
That is, w( x)L is alway a self - adjoint operator.
3
Sturm-Liouville differential equation:
The Sturm-Liouville differential equation is defined as an eigenvalue equation taking
the following self-adjoint form:
d 
d
Lu ( x)   p( x) u ( x)  q( x)u ( x)  w( x)u ( x)
dx 
dx 
 p( x)u" ( x)  p' ( x)u ' ( x)  q( x)u ( x)
Here w( x) is called the weight function , u ( x) is called the eigenfunct ion , and  is called
the eigenvalue .
Examples of Sturm-Liouville differential equations:
1) Legendre' s equation.
2) Bessel' s equation
(1  x 2 )u"2 xu' n(n  1)u  0,  1  x  1.
p ( x)  1  x 2
x 2u" xu' (a 2 x 2  n 2 )u  0, 0  x  a.
p( x)  x
q ( x)  0
q ( x)  n 2 / x
w( x)  1
w( x)  x
  n(n  1)
  a 2
4
Boundary conditions:
Suppose u(x) and v(x) are solutions to the Sturm-Liouville differential equation
d 
du ( x) 
Lu ( x ) 
p
(
x
)
 q( x)u ( x)  w( x)u ( x), then


dx 
dx 

b
a
v*Ludx   v* ( pu ' )' qu dx
b
a
b
b
a
a
b
 v pu '   v ' pu ' dx   v*qudx
*
*
a
b
b
b
b
a
a
a
a
 v* pu '  v* ' pu   (v* ' p)' udx   v*qudx
b
b
a
a
 p(v*u 'v* ' u )   [( pv' )' qv]* udx
b
b
a
a
 p(v*u 'v* ' u )   (Lv)* udx.
b
If we assume to choose the boundary conditions so that p(v*u 'v* ' u)  0 for any two
a
solutions u and v, then the Sturm-Liouville self-adjoint operator satisfies:

b
a
b
v*Ludx   (Lv)* udx.
a
The operator L is then a Hermitian operator. That is

b
b

*
*


v
L
udx

(
L
v
)
udx

L

L

b


a
a
p(v*u 'v* ' u )  0
a

LL
5
Read: Chapter 10: 1
No homework
6
December 7 Sturm-Liouville theory
Weighted inner products:
Suppose we define weighted inner products for our Sturm-Liouville problem as
b
v u   v* ( x)u( x)w( x)dx , then under the afore-mentioned boundary conditions,
a
b L 
L
L
*
*
* L
v
L
udx

(
L
v
)
udx

v
uwdx

v
uwdx

v
u

vu .


a
a
a w
a  w 
w
w
b
b
*
b
That is, L/w is Hermitian with respect to the weighted inner product.
Real eigenvalues and the orthogonality of eigenfunctions:
Suppose we have the Sturm-Liouville differential equation
L i  i w i , because L/w is Hermitian, then
*

L
L
If i  j then i  i ,
*
j i 
j i  0  i   j j i  
w
w

If i  j then j i  0.


We now have the important Sturm-Liouville theory:
(at the chosen boundary conditions):
1)The eigenvalues of a Sturm-Liouville operator are real.
2)The eigenfunctions of a Sturm-Liouville operator are orthogonal with respect to the
weighted inner product.
7
Completeness of eigenfunctions:
The third part of the Sturm-Liouville theory is
3) The eigenfunctions of a Sturm-Liouville operator form a complete set.
This means that any function f (x) satisfying the boundary conditions can be expanded as
f ( x)   an n ( x), where an    n ( x) f ( x) w( x)dx (suppose  n is normalized .)
b
*
a
n
2
m


Mathematically n(x) is complete if lim   f ( x)   ann ( x) w( x)dx  0.
m a
n 0


b
Complete relation of the eigenfunctions:
b
*
f ( x)   an n ( x)      n ( x' ) f ( x' ) w( x' )dx'  n ( x)
a

n
n 
b

*
   w( x' )  n ( x' ) n ( x) f ( x' )dx' 
a
n


adjoint
w( x' )  n ( x' ) n ( x)   ( x  x' ) 
*
n
w( x)  n ( x' ) n ( x)   ( x  x' ) 
*
n
w( x) w( x' )   n ( x' ) n ( x)   ( x  x' )
*
n
8
Example: Fourier series.
Consider the Sturm-Liouville problem
u"u  0, 0  x  L, with the boundary condition u (0)  u ( L) and u ' (0)  u ' ( L).
Here p ( x)  1, q ( x)  0, w( x)  1.
The normalized eigenfunct ions are
2

2
n 2x
2
n 2x
 n 2 
cos
,
sin
, n  1,2,3, with λn  
 .
 n 
L
L
L
L

 L 

1



, with λ0  0.
 0
L
Any function f ( x) satisfying the boundary condition can be expanded as
a0  
n 2x
n2x 
f ( x)     an cos
 bn sin
, with
2 n 1 
L
L 
2 L
n 2x

a

cos
f ( x)dx
n


0
L
L

b  2 L sin n 2x f ( x)dx
 n L 0
L
9
Read: Chapter 10: 2-4
No homework
10
December 9 Green’s function-Eigenfunction expansion
10.5 Green’s function-Eigenfunction expansion
Eigenfunction expansion of Green’s function:
For a nonhomogeneous Sturm-Liouville ODE with a source term
Lu ( x)  w( x)u ( x)  f ( x)
We know that by the Green’s function method, if we find G(x, x') that satisfies
LG ( x, x' )  w( x)G ( x, x' )   ( x  x' )
The solution of the nonhomogeneous equation is then u ( x)   G ( x, x' ) f ( x' )dx'.
Therefore the key point is how to find G(x, x').
We also know that according to the Sturm-Liouville theory, any function can be
expanded by the eigenfunctions of the homogeneous equation. This means that u(x) and
G(x, x') can be expanded by the eigenfunctions.
Let us try to do it.
11
Suppose Lu ( x)  w( x)u ( x)  f ( x), Ln ( x)  n w( x)n ( x), u ( x)   ann ( x), with
n
b
an    n ( x)u ( x) w( x)dx, and  n is normalized .
*
a
Lu ( x)  w( x)u ( x)  f ( x) 

b

b
a
a
b
b
a
a
 n*Ludx     n*uwdx    n* fdx 
b
(L n )* udx an    n fdx 
*
a
b
b
a
a
n   n*uwdx  an    n* fdx 
b
n an  an    n* fdx  an 

b
a
 n* ( x' ) f ( x' )dx'
n  
a

b
*

n ( x ' ) f ( x ' ) dx '

a
u ( x)   ann ( x)  
n ( x) 
n  
n
n

 n* ( x' )n ( x)
u ( x)   
f ( x' )dx'
a

n  
n
 G ( x, x ' ) 
b
b
u ( x)   G ( x, x' ) f ( x' )dx'
a



 n* ( x' )n ( x)
n   
n
12
Eigenfunction expansion of Green’s function: Example: Helmholtz equation
For  2 k (r )  k 2 k (r ) , where L   2 is the Sturm - Liouville operator,  k 2 is the eigenvalue ,
and w(r )  1, we have the palne - wave eigenfunct ions  k (r ) 
1
2 
3/ 2
e ik r , forming a complete
orthonorma l set  k*1 (r ) k 2 (r )d   (k 1  k 2 ).
Now for the nonhomegen eous equation  2  (r )  k02  (r )  f (r ), the Green' s function satisfies

2

 k02 G (r, r ' )   (r  r ' ).
ik r r '
 n* ( x' )n ( x)
 k* (r ' ) k (r )d k
1
e ik r r ' 
e
G ( x, x ' )  
 G (r, r ' )  

d k  
.
3  2
2
2
2



(

k
)

(

k
)
k

k
4

r

r
'


2

n
n
0
0
0
1
 (r )   G (r, r ' ) f (r ' )d '  
4
ik r r '
e 0
 r  r' f (r' )d '.
13
Read: Chapter 10: 5
No homework
14