Mathematical Methods for Physics and Engineering : A Comprehensive Guide

17.7 EXERCISES

We note that ifμ=λn,i.e.ifμequals one of the eigenvalues ofL,thenG(x, z)

becomes infinite and this method runs into difficulty. No solution then exists

unless the RHS of (17.53) satisfies the relation

∫b

a

yˆ∗n(x)f(x)dx=0.

If the spectrum of eigenvalues of the operatorLis anywhere continuous,

the orthonormality and closure relationships of the normalised eigenfunctions

become
∫b

a

yˆn∗(x)yˆm(x)ρ(x)dx=δ(n−m),

∫∞

0

yˆ∗n(z)yˆn(x)ρ(x)dn=δ(x−z).

Repeating the above analysis we then find that the Green’s function is given by

G(x, z)=

∫∞

0

yˆn(x)yˆ∗n(z)

λn−μ

dn.

17.7 Exercises

17.1 By considering〈h|h〉,whereh=f+λgwithλreal, prove that, for two functions
fandg,
〈f|f〉〈g|g〉≥^14 [〈f|g〉+〈g|f〉]^2.

The functiony(x) is real and positive for allx. Its Fourier cosine transform ̃yc(k)

is defined by

̃yc(k)=

∫∞

−∞

y(x)cos(kx)dx,

and it is given thaty ̃c(0) = 1. Prove that

̃yc(2k)≥2[y ̃c(k)]^2 − 1.

17.2 Write the homogeneous Sturm-Liouville eigenvalue equation for whichy(a)=
y(b)=0as
L(y;λ)≡(py′)′+qy+λρy=0,
wherep(x),q(x)andρ(x) are continuously differentiable functions. Show that if
z(x)andF(x)satisfyL(z;λ)=F(x), withz(a)=z(b)=0,then
∫b

a

y(x)F(x)dx=0.

Demonstrate the validity of this general result by direct calculation for the
specific case in whichp(x)=ρ(x)=1,q(x)=0,a=−1,b=1andz(x)=1−x^2.
17.3 Consider the real eigenfunctionsyn(x) of a Sturm–Liouville equation,

(py′)′+qy+λρy=0,a≤x≤b,

in whichp(x),q(x)andρ(x) are continuously differentiable real functions and

p(x) does not change sign ina≤x≤b.Takep(x) as positive throughout the