Mathematical Methods for Physics and Engineering : A Comprehensive Guide

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS

which is a statement of the orthogonality ofyiandyj.

If one (or more) of the eigenvalues is degenerate, however, we have different

eigenfunctions corresponding to the same eigenvalue, and the proof of orthogo-

nality is not so straightforward. Nevertheless, an orthogonal set of eigenfunctions

may be constructed using theGram–Schmidt orthogonalisationmethod mentioned

earlier in this chapter and used in chapter 8 to construct a set of orthogonal

eigenvectors of an Hermitian matrix. We repeat the analysis here for complete-

ness.

Suppose, for the sake of our proof, thatλ 0 isk-fold degenerate, i.e.

Lyi=λ 0 ρyi fori=0, 1 ,...,k− 1 , (17.25)

but thatλ 0 is different from any ofλk,λk+1, etc. Then any linear combination of

theseyiis also an eigenfunction with eigenvalueλ 0 since

Lz≡L

∑k−^1

i=0

ciyi=

∑k−^1

i=0

ciLyi=

∑k−^1

i=0

ciλ 0 ρyi=λ 0 ρz. (17.26)

If theyidefined in (17.25) are not already mutually orthogonal then consider

the new eigenfunctionsziconstructed by the following procedure, in which each

of the new functionsziis to be normalised, to givezˆi, before proceeding to the

construction of the next one (the normalisation can be carried out by dividing

the eigenfunctionziby (

∫b

az

∗

iziρdx)

1 / (^2) ):
z 0 =y 0 ,
z 1 =y 1 −
(
ˆz 0
∫b
a
zˆ∗ 0 y 1 ρdx
)
,
z 2 =y 2 −
(
ˆz 1
∫b
a
zˆ∗ 1 y 2 ρdx
)
−
(
zˆ 0
∫b
a
zˆ 0 ∗y 2 ρdx
)
,
..
.
zk− 1 =yk− 1 −
(
zˆk− 2
∫b
a
ˆzk∗− 2 yk− 1 ρdx
)
−···−
(
ˆz 0
∫b
a
zˆ∗ 0 yk− 1 ρdx
)
.
Each of the integrals is just a number and thus each new functionziis, as can be
shown from (17.26), an eigenvector ofLwith eigenvalueλ 0. It is straightforward
to check that eachziis orthogonal to all its predecessors. Thus, by this explicit
construction we have shown that an orthogonal set of eigenfunctions of an
Hermitian operatorLcan be obtained. Clearly the orthogonal set obtained,zi,is
not unique.
In general, sinceLis linear, the normalisation of its eigenfunctionsyi(x)is
arbitrary. It is often convenient, however, to work in terms of the normalised
eigenfunctionsyˆi(x), so that
∫b
ayˆ
∗
iyˆiρdx= 1. These therefore form an orthonormal