Mathematical Methods for Physics and Engineering : A Comprehensive Guide

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS

17.14 Express the solution of Poisson’s equation in electrostatics,

∇^2 φ(r)=−ρ(r)/ 0 ,
whereρis the non-zero charge density over a finite part of space, in the form of
an integral and hence identify the Green’s function for the∇^2 operator.
17.15 In the quantum-mechanical study of the scattering of a particle by a potential,
a Born-approximation solution can be obtained in terms of a functiony(r)that
satisfies an equation of the form
(−∇^2 −K^2 )y(r)=F(r).
Assuming thatyk(r)=(2π)−^3 /^2 exp(ik·r) is a suitably normalised eigenfunction of
−∇^2 corresponding to eigenvaluek^2 , find a suitable Green’s functionGK(r,r′). By
taking the direction of the vectorr−r′as the polar axis for ak-space integration,
show thatGK(r,r′) can be reduced to
1
4 π^2 |r−r′|

∫∞

−∞

wsinw

w^2 −w^20

dw,

wherew 0 =K|r−r′|.

[ This integral can be evaluated using a contour integration (chapter 24) to give

(4π|r−r′|)−^1 exp(iK|r−r′|). ]

17.8 Hints and answers

17.1 Express the condition〈h|h〉≥0 as a quadratic equation inλand then apply the
condition for no real roots, noting that∫ 〈f|g〉+〈g|f〉is real. To put a limit on
ycos^2 kx dx,setf=y^1 /^2 coskxandg=y^1 /^2 in the inequality.
17.3 Follow an argument similar to that used for proving the reality of the eigenvalues,
but integrate fromx 1 tox 2 , rather than fromatob.Takex 1 andx 2 as two
successive zeros ofym(x) and note that, if the sign ofymisαthen the sign ofym′(x 1 )
isαwhilst that ofy′m(x 2 )is−α. Now assume thatyn(x) does not change sign in
the interval and has a constant signβ; show that this leads to a contradiction
between the signs of the two sides of the identity.
17.5 (a)y=

∑

anPn(x)with

an=

n+1/ 2

b−n(n+1)

∫ 1

− 1

f(z)Pn(z)dz;

(b) 5x^3 =2P 3 (x)+3P 1 (x), givinga 1 =1/4anda 3 = 1, leading toy=5(2x^3 −x)/4.
17.7 (a) No,

∫

gf∗′dx= 0; (b) yes; (c) no,i

∫

f∗gdx=0;(d)yes.
17.9 The normalised eigenfunctions are (2/π)^1 /^2 sinnx,withnan integer.
y(x)=(4/π)

∑

nodd[(−1)

(n−1)/ (^2) sinnx]/[n (^2) (κ−n (^2) )].
17.11 λn=(n+1/2)^2 π^2 ,n=0, 1 , 2 ,....
(a) Sinceyn(1)y′m(1)= 0, the Sturm–Liouville boundary conditions are not satis-
fied and the appropriate weight function has to be justified by inspection. The
normalised eigenfunctions are

√

2 e−x/^2 sin[(n+1/2)πx], withρ(x)=ex.

(b)y(x)=(− 2 /π^3 )

∑∞

n=0e

−x/ (^2) sin[(n+1/2)πx]/(n+1/2) (^3).
17.13 yn(x)=

√

2 x−^1 /^2 sin(nπlnx)withλn=−n^2 π^2 ;

an=

{

−(nπ)−^2

∫e

1

√

2 x−^1 sin(nπlnx)dx=−

√

8(nπ)−^3 fornodd,

0forneven.

17.15 Use the form of Green’s function that is the integral over all eigenvalues of the
‘outer product’ of two eigenfunctions corresponding to the same eigenvalue, but
with argumentsrandr′.