Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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256 7 Sturm–Liouville Theory
Figure 7.5. ∆F
n
x for bead at x L/2 in units where L 1.
or
F
n
*F
0
n
2Μ
Π
Modn, 2
mn
F
0
m
mn/ 2
Modm, 2
m n
(7.234)
and are unchanged for even n. Notice that the admixture of terms with m n is pro-
portional to m n
1
, so that nearby terms contribute most; nevertheless, this expansion
converges relatively slowly.
To illustrate the effects of this perturbation, it is useful to express our result in the form
F
n
x*F
0
n
x
2Μ
Π
∆F
n
x (7.235)
Figure 7.5 displays the lowest several ∆F
n
x in units where L 1. These functions are
qualitatively similar to Green functions with central cusps produced by the small bead, but
exist for k k
0
n
.
7.6 Variational Methods
At the beginning of this chapter we derived the Sturm–Liouville equation using a varia-
tional argument and here we would like to extend this approach to a variational analysis of
its eigenvalues. Consider the functional
0Ψ
b
a
px
Ψx
x
2
qxΨx
2
x
b
a
Ψx
2
wxx
(7.236)
7.6 Variational Methods 257
where Ψ is any continuous function that complies with appropriate self-adjoint boundary
conditions and where px and wx are nonnegative in a x b. This functional is
known as the Rayleigh quotient. Since we have shown that both the real and imaginary
components satisfy the same Sturm–Liouville equation, we may assume without loss of
generality that Ψx is real and not use complex conjugates in the definition of 0; this will
simplify the algebra. First, we demonstrate that functions Ψ that render 0Ψ] stationary
with respect to small variations Ψx!Ψx∆Ψx consistent with the boundary condi-
tions are eigenfunctions of the Sturm–Liouville operator. Recognizing that
0
A
B
∆0
B∆A A∆B
B
2
∆A 0∆B
B
(7.237)
and that B is positive-definite, we require
∆0 0 ∆A 0∆B 0 (7.238)
The variation of the numerator
∆A 2
b
a
px
Ψx
x
d∆Ψx
x
qxΨx∆Ψx
x (7.239)
can be simplified using partial integration
1
2
∆A
px
Ψx
x
∆Ψx
b
a
b
a
x
px
Ψx
x
qxΨx
∆Ψxx (7.240)
and the self-adjoint boundary conditions
px
Ψx
x
∆Ψx
b
a
0 (7.241)
to write
∆A 2
b
a
x
px
Ψx
x
qxΨx
∆Ψxx (7.242)
The variation of the denominator
∆B 2
b
a
Ψx∆Ψxwxx (7.243)
is already simple. Combining these terms now gives
∆A 0∆B 0
b
a
x
px
Ψx
x
qxΨx0wxΨx
∆Ψxx 0
(7.244)
If this integral is to vanish for arbitrary ∆Ψx, the coefficient of ∆Ψx must vanish identi-
cally, such that
x
px
Ψx
x
qxΨx0wxΨx0 (7.245)
258 7 Sturm–Liouville Theory
Thus, if we identify 0Ψ with the eigenvalue Λ we recover the Sturm–Liouville eigenvalue
equation
x
px
Ψ
n
x
x
qxΨ
n
xΛ
n
wxΨ
n
x0 (7.246)
Therefore, if 0Ψ
n
is stationary with respect to small variations Ψ
n
!Ψ
n
∆Ψthen Ψ
n
x
is an eigenfunction of the Sturm–Liouville operator with eigenvalue Λ
n
0Ψ
n
.
Suppose that
Ψx
n1
c
n
F
n
x (7.247)
is expanded in a complete orthonormal set of eigenfunctions F
n
satisfying
F
m
w F
n
b
a
F
m
xF
n
xwxx ∆
m,n
(7.248)
The numerator of the Rayleigh quotient then takes the form
b
a
px
Ψx
x
2
qxΨx
2
x
m,n
c
m
c
n
b
a
pxF
'
m
xF
'
n
xqxF
m
xF
n
x x (7.249)
and can be simplified using partial integration of the first term and the boundary conditions
to obtain
0
n
c
2
n
m,n
c
m
c
n
b
a
F
m
x
x
pxF
'
n
x qxF
n
x
x (7.250)
The term in parentheses can be simplified further using the Sturm–Liouville eigenvalue
equation, leaving
0
n
c
2
n
m,n
c
m
c
n
b
a
F
m
xΛ
n
wxF
n
x x (7.251)
Thus, we find that
0Ψ
n
c
2
n
Λ
n
n
c
2
n
Ψ Ψ
Ψw Ψ
(7.252)
where
x
px
x
qx (7.253)
is the Sturm–Liouville operator. Therefore, the Rayleigh quotient can be interpreted as the
average of the eigenvalues weighted by the intensity of the corresponding eigenfunction
7.6 Variational Methods 259
in Ψ. Here intensity is interpreted as the square of the expansion coefficient and represents
the contribution of that term to the normalization integral for Ψ, as given by Parseval’s the-
orem. This result should be familiar from Fourier analysis but applies to any self-adjoint .
Our analysis of the interleaving of roots demonstrated that there exists a smallest eigen-
value; hence, the smallest eigenvalue must represent an absolute minimum for 0Ψ. Sup-
pose that Ψx is a trial function that is intended to approximate F
1
x. The trial function
can be formally expanded in terms of eigenfunctions as
ΨxF
1
x
n2
a
n
F
n
x (7.254)
where each a
n
is small if Ψ is a decent approximation to F
1
. Notice that we are free to use
F
1
w Ψ 1 (7.255)
because 0Ψ is independent of normalization. Then
0Ψ
Λ
1
n2
a
2
n
Λ
n
1
n2
a
2
n
*Λ
1
n2
a
2
n
Λ
n
Λ
1
(7.256)
demonstrates that the error in the approximation to Λ
1
due to the error (ΨF
1
) is second-
order with respect to a
n
. Thus, even if the trial function is not especially good, the estimate
Λ
1
*0Ψmay still be pretty accurate. Furthermore, the true value is smaller than that
obtained with any trial function (other than F
1
itself). Therefore, if we construct a trial
function that resembles F
1
based upon intuition and which includes a parameter, mini-
mization of 0Ψ with respect to that parameter should produce a good approximation
to Λ
1
. Naturally, a better trial function provides a better estimate of Λ
1
. This procedure is
known as the Rayleigh–Ritz method.
7.6.1 Example: Vibrating String
The almost universal first example of this technique is the vibrating string. Although we
already know the eigenvalues and eigenfunctions for
2
x
2
k
2
n
F
n
0 , F
n
0F
n
L0 (7.257)
with Rayleigh quotient
0Ψ
L
0
Ψ
'
x
2
x
L
0
Ψx
2
x
(7.258)
suppose that we feign ignorance and employ a zero-parameter trial solution of the form
ΨxxL x (7.259)
260 7 Sturm–Liouville Theory
which satisfies the boundary conditions and is symmetric with respect to L/2, as expected
for the lowest eigenfunction. Thus,
Ψ
'
xL 2x 0Ψ
L
0
L 2x
2
x
L
0
x
2
L x
2
x
10
L
2
(7.260)
is only 1.3 % larger than the exact answer of Π
2
/L
2
. A better approximation can be obtained
by including a parameter. Consider, for example, a trial function of the form
Ψx_xL x
1 Α
x
L
2
2
Although the evaluation of 0 is straightforward, we will use to perform that
unenlightening task
Integrate
Ψ
!
x
2
, x, 0, L
Integrate
Ψx
2
, x, 0, L
//Simpli fy
6
560 56L
2
Α11L
4
Α
2
L
2
336 24L
2
ΑL
4
Α
2
and then minimize 0 with respect to the parameter Α, to obtain
sol Solve
"
Α
0, Α
Α
4
49 4
133
13L
2
,
Α
4
49 4
133
13L
2
Π
2
/ .sol / / N
1.00001
L
2
,
10.348
L
2
Obviously one of the solutions corresponds to a maximum and should be rejected, while
the other brings us within one part in 10
5
of the exact answer. Of course, a poor choice of
trial function, such as the addition of a term that is odd with respect to the midpoint, would
bring little or no improvement in the variational estimate. By respecting the symmetry of
the problem, this judicious choice of trial function offers a very accurate estimate.
Problems for Chapter 7
1. Laguerre polynomials
Suppose that L
n
x is a polynomial of order n and that polynomials of different order are
orthogonal with respect to the exponential weight function wx
x
on the interval
0 x<, such that
0
L
n
xL
m
x
x
x ∆
n,m
(7.261)
Problems for Chapter 7 261
a) Construct the lowest three Laguerre polynomials using orthonormality conditions.
b) Alternatively, show that these functions satisfy a Sturm–Liouville equation that can
be expressed in the form
xy
''
xgxy
'
xΛ
n
yx0 (7.262)
and demonstrate that the eigenvalue must be a positive integer for the corresponding
eigenfunction to be bounded as x !.
2. Minimization of mean-square error
Suppose that a piecewise smooth function f x is approximated by the truncated expansion
f
N
x
N
m0
c
m
F
m
x (7.263)
where the eigenfunctions F
m
are orthonormal with respect to the weight function wx on
the interval a x b, such that
b
a
F
m
x
F
n
xwxx ∆
m,n
(7.264)
Show that the mean-square error
∆
2
N
b
a
f x f
N
x
2
wxx (7.265)
is minimized when
c
m
b
a
F
m
x
f xwxx (7.266)
independent of N. Note that this independence of the optimum coefficients c
m
from the
number of terms N is a special property of eigenfunction expansions that would not be
satisfied if the coefficients of arbitrary functions, such as x
m
, were determined by least-
squares fitting. (Hint: consider independent variations of c
m
and c
m
.)
3. Intermediate boundary conditions
Show that a regular Sturm–Liouville operator (with pa0 and pb0) is self-adjoint
in a x b for any nontrivial intermediate boundary conditions of the form
Α
0
f aΑ
1
f
'
a0 (7.267)
Β
0
f bΒ
1
f
'
b0 (7.268)
where the coefficients Α
i
and Β
i
for i 0, 1 are real.
262 7 Sturm–Liouville Theory
4. Eigenfunctions for string with intermediate boundary conditions
The eigenfunctions for a vibrating string with intermediate boundary conditions satisfy
2
x
2
k
2
n
F
n
x0 (7.269)
Α
0
F
n
0Α
1
F
'
n
00 (7.270)
Β
0
F
n
LΒ
1
F
'
n
L0 (7.271)
Derive a transcendental equation for the eigenvalues k
n
and discuss a graphical procedure
for determining numerical values.
5. Sturm–Liouville Wronskian
Suppose that y
1
and y
2
are two solutions to the Sturm–Liouville equations
Λwxy
1
x0 (7.272)
Λwxy
2
x0 (7.273)
where
x
px
x
qx (7.274)
Evaluate their Wronskian
Wxy
1
xy
'
2
xy
2
xy
'
1
x (7.275)
and show that pxWx is constant. (Hint: evaluate W
'
x.)
6. Bounds on eigenvalues
Consider the eigenvalue problem
1 x
2
y
''
x2xy
'
xΛ1 x
2
yx0 (7.276)
y
'
00 ,y10 (7.277)
in the interval 0 x 1. Show that the lowest eigenvalue is in the range
Π
2
8
Λ
1
Π
2
2
.
7. Green function for vibrating string
The Green function for a vibrating string clamped at both ends satisfies
2
x
2
k
2
G
k
x, x
'
∆x x
'
,G
k
0,x
'
G
k
,x
'
0 (7.278)
a) Write down the formal eigenfunction expansion of G
k
x, x
'
. Under what conditions
does the Green function exist?
b) Demonstrate that the Green function can also be expressed in closed form as
G
k
x, x
'
Sinkx
<
Sinkx
>
k Sink
(7.279)
where x
<
is the smaller and x
>
is the larger of x and x
'
.
Problems for Chapter 7 263
c) Prove that these representations are equivalent.
8. Green function for y
''
k
2
y f x with y0yL0
Consider the differential equation
y
''
xk
2
yx f x ,y0yL0 (7.280)
where k>0.
a) Construct a closed form for the Green function and write a general form for the solu-
tion to the inhomogeneous equation.
b) Construct the Green function as a Fourier sine series and again write a general form
for the solution to the inhomogeneous equation.
c) Demonstrate that these two representations are equivalent.
9. Green function for stopped pipe
Acoustical vibrations in a stopped pipe satisfy
y
''
xk
2
yx f x ,y
'
00 ,yL0 (7.281)
a) Construct a closed form for the Green function and write a general form for the solu-
tion of the inhomogeneous equation.
b) Construct the Green function as an eigenfunction series and again write a general form
for the solution of the inhomogeneous equation.
c) Demonstrate that these representations are equivalent.
10. Green function for open pipe
Acoustical vibrations in an open pipe satisfy
y
''
xk
2
yx f x ,y
'
0y
'
L0 (7.282)
a) Construct a closed form for the Green function and write a general form for the solu-
tion of the inhomogeneous equation.
b) Construct the Green function as an eigenfunction series and again write a general form
for the solution of the inhomogeneous equation.
c) Demonstrate that these representations are equivalent.
11. Green function for periodic orbits
Suppose that small-amplitude oscillations around a stable orbit in a synchrotron satisfy a
differential equation of the form
y
''
xΩ
2
yx f x , 0 x 2Π (7.283)
subject to periodic boundary conditions
y0y2Π ,y
'
0y
'
2Π (7.284)
where f x is a localized disturbance.
264 7 Sturm–Liouville Theory
a) Obtain the eigenvalues Ω
m
and eigenfunctions F
m
x for the homogeneous equation
and write a formal expansion for its Green function.
b) Obtain an explicit closed-form solution for the Green function.
c) Write a formal expression for the solution to the inhomogeneous equation.
d) Discuss the behavior of the system near a resonance with Ω*Ω
m
.
12. Fredholm alternative theorem
Suppose that you have in your possession the eigenvalues and eigenvectors Λ
m
, F
m
x that
satisfy a homogeneous Sturm–Liouville problem of the form
Λ
m
wxF
m
x0 ,
x
px
x
qx (7.285)
and wish to solve the inhomogeneous equation
Λ
m
wxΨx f x (7.286)
by expanding Ψ in terms of F
m
. The boundary conditions are homogeneous but for our
present purposes need not be specified further. What condition must f satisfy for this
strategy to succeed? Write a formal solution assuming that f satisfies your condition. What
happens if f fails to satisfy this condition? The general solution to this problem is called
the Fredholm alternative theorem.
13. Green function for a 4th order equation
Consider a linear fourth-order differential equation of the form
4
x
4
Λ
yx0 ,y0y
''
0yΠ y
''
Π 0 (7.287)
a) Show that this equation is self-adjoint for the specified boundary conditions.
b) Obtain the normalized eigenfunctions Φ
n
x for 0 x Πand corresponding eigen-
values Λ
n
.
c) Construct an eigenfunction expansion for the Green function that satisfies
d
4
x
4
Λ
G
Λ
x, x
'
∆x x
'
(7.288)
with the same boundary conditions.
d) Use the Green function to produce a series solution to the inhomogeneous equation
d
4
x
4
Λ
yxx, y0y
''
0yΠ y
''
Π 0 (7.289)
Problems for Chapter 7 265
14. Shifting the eigenvalue spectrum
Suppose that
1
x
p
1
x
x
q
1
x
1
Α
n
w
1
xΨ
n
x0 (7.290)
has an eigenvalue spectrum Α
n
,n 0, with Α
0
0. Construct a related Sturm–
Liouville operator
2
x
p
2
x
x
q
2
x
2
Β
n
w
2
xΨ
n
x0 (7.291)
that has the same eigenfunctions and a spectrum of eigenvalues Β
n
that is shifted with
respect to Α
n
such that Β
0
0. How are p
2
,q
2
,w
2
related to p
1
,q
1
,w
1
?
15. An eigenvalue problem with mixed boundary conditions
Show that the eigenvalue problem
y
''
Λy 0 ,y00 ,y
'
0y1 (7.292)
with mixed boundary conditions has only one real eigenvalue and determine the corre-
sponding eigenfunction. Why does this problem not have an denumerably infinite spec-
trum of solutions?
16. Normal modes for stretched string with one end attached to spring
Suppose that a string of length L under tension T has one end fixed while the other is
attached to a ring that slides without friction on a fixed rod and that the ring is attached to
a spring with spring constant Α. Thus, the transverse displacement yx,t satisfies
Ρ
(
2
yx, t
(t
2
T
(
2
yx, t
(x
2
0 ,y0,t0 ,
(yx, t
(x
xL
Α
T
yL, t (7.293)
where Ρ is the mass density.
a) Show that there exist normal modes of the form
yx, t
Ω
n
t
f k
n
x (7.294)
and determine the relationship between Ω
n
and the parameters of the problem L, T,
Α, Ρ. Discuss the special cases of very weak and very strong springs.
b) Demonstrate that the normal modes are orthogonal.
17. Hanging string
A string of length L with uniform mass density Σ hangs under its own weight. Here we
investigate small-amplitude transverse vibrations in a plane, assuming that the tension Τx
is not affected by this motion.
a) Evaluate the equilibrium tension Τx and construct the normal-mode equation. What
are the appropriate boundary conditions?