Mathematical Methods for Physics and Engineering : A Comprehensive Guide

17.5 SUPERPOSITION OF EIGENFUNCTIONS: GREEN’S FUNCTIONS

17.5 Superposition of eigenfunctions: Green’s functions

We have already seen that if

Lyn(x)=λnρ(x)yn(x), (17.43)

whereLis an Hermitian operator, then the eigenvaluesλnare real and the

eigenfunctionsyn(x) are orthogonal (or can be made so). Let us assume that we

know the eigenfunctionsyn(x)ofLthat individually satisfy (17.43) and some

imposed boundary conditions (for whichLis Hermitian).

Now let us suppose we wish to solve the inhomogeneous differential equation

Ly(x)=f(x), (17.44)

subject to the same boundary conditions. Since the eigenfunctions ofLform a

complete set, the full solution,y(x), to (17.44) may be written as a superposition

of eigenfunctions, i.e.

y(x)=

∑∞

n=0

cnyn(x), (17.45)

for some choice of the constantscn. Making full use of the linearity ofL, we have

f(x)=Ly(x)=L

(∞

∑

n=0

cnyn(x)

)

=

∑∞

n=0

cnLyn(x)=

∑∞

n=0

cnλnρ(x)yn(x).

(17.46)

Multiplying the first and last terms of (17.46) byy∗jand integrating, we obtain

∫b

a

y∗j(z)f(z)dz=

∑∞

n=0

∫b

a

cnλny∗j(z)yn(z)ρ(z)dz, (17.47)

where we have usedzas the integration variable for later convenience. Finally,

using the orthogonality condition (17.27), we see that the integrals on the RHS

are zero unlessn=j, and so obtain

cn=

1

λn

∫b

ay

∗

n(z)f(z)dz

∫b

ay

∗

n(z)yn(z)ρ(z)dz

. (17.48)

Thus, if we can find all the eigenfunctions of a differential operator then (17.48)

can be used to find the weighting coefficients for the superposition, to give as the

full solution

y(x)=

∑∞

n=0

1

λn

∫b

ay

∗

n(z)f(z)dz

∫b

ay

∗

n(z)yn(z)ρ(z)dz

yn(x). (17.49)

If we work with normalised eigenfunctionsyˆn(x), so that

∫b

a

yˆn∗(z)yˆn(z)ρ(z)dz= 1 for alln,