Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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226 7 Sturm–Liouville Theory
The general string equation is one example of a Sturm–Liouville eigenvalue problem.
Let
x
px
x
qxΨ
x
px
Ψx
x
qxΨx (7.25)
represent a linear differential operator based upon the coefficient functions px and qx.
Based upon the mechanical problem discussed above, we could describe px as a force
density and qx as an inertial density. An eigenvalue equation for a system with weight or
metric function wx that is analogous to the mass density then takes the form
ΛwΨ 0
x
px
Ψx
x
qxΨxΛwxΨx0 (7.26)
where one expects nontrivial solutions, Ψ
n
, for discrete eigenvalues, Λ!Λ
n
, when suitable
boundary conditions are applied for a finite interval. Many physics problems can be rep-
resented in this manner. Therefore, in this chapter we shall study the general properties of
one-dimensional Sturm–Liouville systems and the application of eigenvalue expansions to
the solution of inhomogeneous equations of the general form
ΛwΨx f x (7.27)
The chapter on boundary-value problems will later apply these techniques to problems
in two or three spatial dimensions using separable coordinate systems. After separation
of variables one generally obtains Sturm–Liouville systems for each spatial dependence
and these systems are connected by means of separation constants that depend upon the
eigenvalues for the other equations.
7.2 Hilbert Spaces
In the broadest sense a Hilbert space is any infinite-dimensional linear vector space with an
inner product. An abstract linear vector space is a set of elements (vectors) with addition
and multiplication operations that satisfy the following properties.
1. closure under addition: 9 f,g ,h f g
2. commutativity of addition: f g g f
3. associativity of addition: f g hf gh
4. existence of additive identity: : 0 % f 0 f
5. closure under multiplication by scalar: 9 f , Α f
6. commutativity of multiplication by scalar: ΑΒf ΑΒ f
7. distributive law: Α f gΑf Αg
8. associative law: Α Β f Αf Βf
7.2 Hilbert Spaces 227
9. existence of multiplicative identity: : 1 % 1 f f.
10. existence of additive inverse: 9 f : g % f g 0.
The inner product f gfor a real Hilbert space must satisfy:
1. f g g f
2. f g h f g f h
3. f f 2 0 with equality iff f 0,
while for a complex Hilbert space we require:
1. f g g f
2. f g h f g f h
3. f f 2 0 with equality iff f 0.
A linear Hermitian operator acting upon the elements f,g is defined by the
requirements
g f g (7.28)
Α f ΒgΑf Βg (7.29)
f g g f
f g (7.30)
where Α and Β are arbitrary constants. The complex conjugation properties are often com-
bined in the statement
(7.31)
where
is the Hermitian adjoint of . For an ordinary vector space, is a square matrix
and
is the complex conjugate of its transpose; hence, a Hermitian matrix is identical to
its Hermitian adjoint and is described as self-adjoint. The notation
f g f g f g (7.32)
expresses the fact that a Hermitian can act to the right or its adjoint
can act to the left
without changing the value of the matrix element.
Two elements of are described as orthogonal when their inner product
f ]g f g 0 (7.33)
vanishes. It is often useful to generalize the notion of orthogonality to include a metric
function w, where in a conventional vector space w is a positive-definite diagonal matrix,
228 7 Sturm–Liouville Theory
meaning a diagonal matrix whose elements are real, positive, and nonzero. A matrix of
this type is clearly self-adjoint, such that
f w g
g w f w w
(7.34)
satisfies the complex conjugate property required for the inner product of a Hilbert space.
Thus, two elements are described as orthogonal with respect to w when
f ]g f w g 0 (7.35)
This notion of orthogonality with respect to a metric function will prove useful in more
general Hilbert spaces also.
For the purposes of this chapter we define a particular type of Hilbert space consisting
of functions of a single real variable that are smooth and continuous on a finite interval
a x b and satisfy boundary conditions of the form
a
f
b
f (7.36)
where
a
n
j1
Α
j
(
(x
j
,
b
n
j1
Β
j
(
(x
j
(7.37)
are linear differential operators evaluated at the endpoints of the domain. Hence, we require
the elements f x to possess at least n continuous derivatives in the interior of the
domain. We will refer to real Hilbert spaces with boundary conditions of this type as
Sturm–Liouville spaces and complex Hilbert spaces of this type as Hermitian spaces.The
simplest boundary conditions can be further classified as
Dirichlet: faf b0 (7.38)
Neumann: f
'
a f
'
b0 (7.39)
periodic: f a f b,f
'
a f
'
b (7.40)
More generally, boundary conditions with f 0 are described as homogeneous.
Similarly, we define the inner product for a real function space as
f g
b
a
f xgxx g f (7.41)
while for a complex function space we define
f g
b
a
f
xgxx g f
(7.42)
Notice that these definitions implicitly impose another constraint on – the existence of
an inner product requires the elements of to be square integrable. Two elements of are
described as orthogonal with respect to the metric function or weight wx when
f ]g f w g 0 (7.43)
where wx20 is real and nonnegative within the domain of . In fact, we generally
assume that wx is free of interior zeros. Usually the requirement of square integrability
7.2 Hilbert Spaces 229
is less demanding with than without the metric function. When using a nontrivial metric
function, we often refer to the matrix element
f w g
b
a
f xgxwxx g w f (7.44)
for real or
f w g
b
a
f
xgxwxx g w f
(7.45)
for complex functions as the inner product, leaving the phrase “with respect to w” under-
stood. For the general string equation where wx is the mass density Σx, we interpret the
combination Σxx m as the mass of a differential element of string. The product of
two functions f x and gx is naturally weighted by the mass m carried by an infinitesi-
mal length x.
A linear differential operator of order n is defined by
n
j1
d
j
x
(
(x
j
(7.46)
where each d
j
x is a fixed function of x that is independent of the element of to which it
is applied. We assume that the functions d
j
x are finite and continuous within the domain
of .Adifferential operator is self-adjoint in a Sturm–Liouville space when
b
a
f xgx x
b
a
gxf x x f g g f (7.47)
or in a Hermitian space when
b
a
f
xgx x
b
a
g
xf x x
f g g f
(7.48)
such that can be shifted from one side to the other without changing the value of the
matrix element.
7.2.1 Schwartz Inequality
The Schwartz inequality
f w g
2
f w f g w g (7.49)
where equality pertains iff f g should be familiar from linear algebra and applies to any
Hilbert space, but we review the derivation to be sure. The theorem is obviously true for
230 7 Sturm–Liouville Theory
the trivial case in which either vector is null, so we assume that neither is. Let h af bg
where a, b are arbitrary scalars, such that
h af bg h w h a
af w f ab
f w g a
bg w f
b
bg w g 2 0 (7.50)
If we choose
a g w g g w gf w f b
f w g bg w f b
b 2 0 (7.51)
and then choose
b f w g g w gf w f g w f f w g2 0 (7.52)
we obtain the Schwartz inequality.
7.2.2 Gram–Schmidt Orthogonalization
Suppose that we possess a set of m linearly independent but possibly nonorthogonal vec-
tors u
j
,j 1,m and wish to construct orthogonal linear combinations
Φ
i
m
j1
a
i, j
u
j
(7.53)
such that
i j
Φ
i
w Φ
j
0 (7.54)
A simple procedure constructs the set Φ
i
one at a time, subtracting off the projections
upon previous vectors. Choose
Φ
1
u
1
(7.55)
and then assign
Φ
2
u
2
Φ
1
w u
2
Φ
1
w Φ
1
Φ
1
(7.56)
such that
Φ
1
w Φ
2
u
1
w u
2
u
1
w u
2
u
1
w u
1
u
1
w u
1
0 (7.57)
Continuing in this manner
Φ
3
u
3
Φ
1
w u
3
Φ
1
w Φ
1
Φ
1
Φ
2
w u
3
Φ
2
w Φ
2
Φ
2
(7.58)
7.2 Hilbert Spaces 231
ensures
Φ
1
w Φ
3
u
1
w u
3
u
1
w u
3
Φ
1
w Φ
1
Φ
1
w Φ
1
Φ
2
w u
3
Φ
2
w Φ
2
Φ
1
w Φ
2
0
(7.59)
Φ
2
w Φ
3
Φ
2
w u
3
u
1
w u
3
Φ
1
w Φ
1
Φ
2
w Φ
1
Φ
2
w u
3
Φ
2
w Φ
2
Φ
2
w Φ
2
0
(7.60)
This procedure can now be repeated enough times to produce m independent orthogonal
vectors of the form
Φ
1
u
1
, Φ
i1
u
i1
i
j1
Φ
j
w u
i1
Φ
j
w Φ
j
Φ
j
(7.61)
Usually we prefer an orthonormal basis, where the vectors are both mutually orthogonal
and normalized, such that
F
i
e
i∆
i
Φ
i
w Φ
i
1
Φ
i
(7.62)
where ∆
i
are real phases which may be chosen at our convenience.
7.2.2.1 Example: Legendre Polynomials
Suppose that we wish to construct a set of polynomials Φ
n
x on the interval 1, 1 that
are orthogonal with respect to the inner product
f g
1
1
f x gxx (7.63)
We can start with a nonorthogonal basis u
n
x
n
with n 2 0 and apply the Gram–Schmidt
orthogonalization procedure
Φ
0
1 , Φ
0
Φ
0
2 (7.64)
Φ
1
x
Φ
0
u
1
Φ
0
Φ
0
Φ
0
x
1
2
1
1
t t x, Φ
1
Φ
1
1
1
t
2
t
2
3
(7.65)
Φ
2
x
2
Φ
0
u
2
Φ
0
Φ
0
Φ
0
Φ
1
u
2
Φ
1
Φ
1
Φ
1
x
2
1
2
1
1
t
2
t x
1
1
t
3
t x
2
1
3
(7.66)
to generate as many terms as our patience permits. Obviously, this is a job for a machine.
TableLegendrePn, x, n, 0, 5
1, x,
1
2
3x
2
2
,
3x
2
5x
3
2
,
3
8
15x
2
4
35x
4
8
,
15x
8
35x
3
4
63x
5
8
The Legendre polynomials are normally defined using the conventional normalization
P
n
11.
232 7 Sturm–Liouville Theory
7.3 Properties of Sturm–Liouville Systems
7.3.1 Self-Adjointness
An important example of a self-adjoint operator is the Sturm–Liouville operator
x
px
x
qxf xpxf
''
xp
'
x f
'
xqx f x (7.67)
where px and qx are real functions and where we can assume, without loss of generality,
that px is nonnegative. Suppose that f x and gx are two piecewise smooth functions
that both satisfy specific boundary conditions at x a, b but are otherwise arbitrary. Such
an operator is self-adjoint whenever these boundary conditions ensure that
b
a
f x
gxx
b
a
gx
f xx
f
p
g
x
b
a
gp
f
x
b
a
(7.68)
(This result is obtained by integrating both sides by parts.) Thus, we require boundary
conditions which will ensure
pb f b
g
'
bgb f
'
b
paf a
g
'
aga f
'
a
(7.69)
for any f,g . Boundary conditions which ensure that a differential operator is self-
adjoint are sometimes described as self-adjoint boundary conditions.
First, consider a regular Sturm–Liouville system for which px is strictly positive
throughout the interval, having no interior zeros and being nonzero at both endpoints.
Perhaps the simplest boundary conditions with this property are the Dirichlet conditions
f a f b0. Also simple are the Neumann conditions f
'
a f
'
b0. More
generally, one can show that any linear conditions of the form
a
f Α
0
f aΑ
1
f
'
a0 (7.70)
b
f Β
0
f bΒ
1
f
'
b0 (7.71)
where Α
j
and Β
j
are fixed real constants (independent of f ), will suffice to ensure that
is self-adjoint. Such conditions are described as unmixed because each equation only
involves values at one endpoint and as homogeneous because
a
f
b
f 0. A bound-
ary condition that involves both a value and a derivative at the same point is described as
intermediate (between Dirichlet and Neumann). When papb,theperiodic boundary
conditions
f a f b ,f
'
a f
'
b (7.72)
will also produce a self-adjoint system. Periodic boundary conditions compare values at
the two endpoints and may be described as mixed. Sometimes it is possible to employ more
complicated mixed boundary conditions, but we will not.
Next, suppose that px vanishes at one or both of the endpoints. At first glance it might
appear that one need not impose boundary conditions upon the elements of , but since
7.3 Properties of Sturm–Liouville Systems 233
usually arises in connection with a differential equation of the form
f xΛwx f x0 (7.73)
we must recognize that a zero of px is a singular point of the differential equation and
that conditions, such as finiteness, will be needed to handle the singularities. If px were
to display interior roots, we would have to have to apply Sturm–Liouville methods to the
intervening regions separately and hope that some method could be devised to bridge the
singular points. However, space does not permit a more general exploration of irregular
Sturm–Liouville problems.
Similar methods can often be used to demonstrate that an operator is self-adjoint on an
infinite domain, either 0 x< or <x<. Under these conditions one usually
needs p ! 0asx !and the appropriate boundary conditions require the solution to
remain finite in order to ensure that the integrated terms vanish and that the operator is
self-adjoint. Similarly, one can extend many of the properties developed here to operators
that satisfy
b
a
f
rg
r V
b
a
g
rf
r V
(7.74)
in two or more dimensions. The integrated terms encountered in the demonstration that the
inner product is self-adjoint with respect to suitable boundary conditions are then described
as surface terms. If the surface terms must vanish for any pair of functions f,g that satisfy
the boundary conditions, then is self-adjoint with respect to those boundary conditions.
The Sturm–Liouville operator is just one example, albeit an important one, of a self-
adjoint operator. A Sturm–Liouville system is defined by its operator, , or equivalently
its coefficient functions px and qx, its boundary conditions
a
and
b
, and its weight
function wx. Many of the important special functions in physics originate in Sturm–
Liouville systems.
7.3.2 Reality of Eigenvalues and Orthogonality of Eigenfunctions
An eigenvalue problem is defined by
u
i
xΛ
i
wxu
i
x0 (7.75)
where is a self-adjoint operator and u
i
satisfies self-adjoint boundary conditions
at the endpoints of a, b. Generally one can satisfy both boundary conditions simultane-
ously only for very particular values of Λ
i
, called characteristic values or eigenvalues.The
corresponding solution u
i
is then called a characteristic function or a normal mode or an
eigenfunction. Consider two eigenfunctions satisfying
u
i
xΛ
i
wxu
i
x0 u
j
x
u
i
xΛ
i
wxu
j
x
u
i
x (7.76)
u
j
xΛ
j
wxu
j
x0 u
i
x
u
j
xΛ
j
wxu
i
x
u
j
x (7.77)
234 7 Sturm–Liouville Theory
such that
b
a
u
j
x
u
i
xx
b
a
u
i
x
u
j
xx
Λ
i
b
a
u
j
x
u
i
xwxx
Λ
j
b
a
u
i
x
u
j
xwxx
Λ
j
Λ
i
b
a
u
j
x
u
i
xwxx (7.78)
where the left-hand side vanishes for any self-adjoint and where we use the fact that the
weight w is real. Thus, recognizing that the normalization integral
Φ
i
w Φ
i
b
a
u
i
x
u
i
xwxx 2 0 (7.79)
is positive-definite for any nontrivial (i.e., nonzero) eigenfunction, we find that the eigen-
values must be real, such that
i j Λ
i
Λ
i
Λ
i
(7.80)
Furthermore, if Λ
i
Λ
j
the integral on the right-hand side must vanish, such that
Λ
i
Λ
j
u
j
w u
i
0
b
a
u
j
x
u
i
xwxx (7.81)
Therefore, eigenfunctions with different eigenvalues are orthogonal with respect to the
weight function. On the other hand, if there are m linearly independent eigenfunctions
with the same eigenvalue, which is described as m-fold degenerate, it is possible to form m
independent linear combinations of the form
Φ
i
x
m
j1
a
i, j
u
j
x (7.82)
that are mutually orthogonal (with respect to w), such that
i j Φ
i
w Φ
j
0 (7.83)
using the Gram–Schmidt orthogonalization procedure. The eigenvalues can then be obtain-
ed from
Λ
i
Φ
i
Φ
i
Φ
i
w Φ
i
(7.84)
where it is not necessary to normalize the eigenfunctions. On the other hand, it is always
possible and usually more convenient to construct an orthonormal basis satisfying
F
i
xΛ
i
wxF
i
x0 , F
i
w F
j
∆
i, j
(7.85)
Note that it is sometimes convenient to distinguish between the orthonormal set F
i
and
more primitive sets Φ
i
before normalization and u
i
before orthogonalization within
degenerate subspaces.
7.3 Properties of Sturm–Liouville Systems 235
7.3.3 Discreteness of Eigenvalues
One can also show that the set of eigenvalues Λ
i
is infinite, bounded from below, and
has at most one accumulation point at . Thus, one can arrange the eigenvalues in an
increasing sequence Λ
0
< Λ
1
< Λ
2
,,, where we omit duplicated values. Often some of the
eigenvalues will be duplicated several times; a value that is repeated m times is described
as m-fold degenerate and then corresponds to m linearly independent eigenfunctions. We
normally assume that the Gram–Schmidt orthogonalization procedure has been used to
construct m linearly independent, mutually orthogonal eigenvectors for each degenerate
eigenvalue. Degeneracies often result from a symmetry, such as reflection or rotational
symmetry.
7.3.4 Completeness of Eigenfunctions
Suppose that Ψx is an arbitrary piecewise smooth function in a, b with a finite number
of discontinuities and that it satisfies the boundary conditions imposed upon a self-adjoint
operator .LetF
n
represent the orthonormal eigenfunctions of , such that
F
n
Λ
n
wF
n
0 (7.86)
F
n
w F
m
b
a
F
n
x
F
m
xwxx ∆
n,m
(7.87)
where wx is the weight function for this system. We can then expand
Ψx
n
a
n
F
n
x (7.88)
a
n
F
n
w Ψ
b
a
F
n
x
Ψxwxx (7.89)
to any desired degree of accuracy. More rigorously, we define the mean-square error in a
truncated expansion with N terms as
$
N
Ψ
b
a
Ψx
N
n1
a
n
F
n
x
2
wxx (7.90)
One can show that the mean-square error approaches zero as the number of terms increases,
such that
lim
N!
$
N
Ψ0 (7.91)
This condition is described as convergence in the mean and is less restrictive than uniform
convergence, but it suffices for practically any application in physics. Indeed, we cannot
expect uniform convergence if Ψ has a discontinuity; the error in the eigenfunction expan-
sion at a discontinuity is sometimes called the Gibb’s phenomenon. Note that uniform
convergence can be established for analytic functions Ψ by comparison with a Laurent
series and expansion of F
n
in power series.