Thus, provided p0 does not vanish, there is always some inner product with

respect to which a real second-order di¬erential operator is formally self-

adjoint.

Note that with

1

Ly = (wp0y ) + p2 y, (4.33)

w

4.2. THE ADJOINT OPERATOR 89

the eigenvalue equation

Ly = »y (4.34)

can be written

(wp0 y ) + p2 wy = »wy. (4.35)

When you come across a di¬erential equation where, in the term containing

the eigenvalue », the eigenfunction is being multiplied by some other function,

you should immediately suspect that the operator will turn out to be self-

adjoint with respect to the inner product having this other function as its

weight.

Illustration (Bargmann-Fock space): This is a more exotic example of a

formal adjoint, although you may have met with it in a course on quantum

mechanics. Consider the space of polynomials P (z) in the complex variable

z = x + iy. De¬ne an inner product by

1 —

d2 z e’z z [P (z)]— Q(z),

P, Q =

π

where d2 z ≡ dx dy and the integration is over the entire x, y plane. With

this inner product, we have

z n , z m = n!δnm .

If we de¬ne

d

a=

ˆ ,

dz

then

1 d

—

d2 z e’z z [P (z)]—

P, a Q

ˆ = Q(z)

π dz

1 d ’z — z

[P (z)]— Q(z)

d2 z

=’ e

π dz

1 —

d2 z e’z z z — [P (z)]— Q(z)

=

π

1 —

d2 z e’z z [zP (z)]— Q(z)

=

π

ˆ

= a† P, Q

ˆ

where a† = z, i.e. the operation of multiplication by z. In this case, the

ˆ

adjoint is not even a di¬erential operator2 .

2

In deriving this result we have observed that z and z — can be treated as independent

90 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS

4.2.2 A Simple Eigenvalue Problem

A ¬nite Hermitian matrix has a complete set of orthonormal eigenvectors.

Does the same property hold for a Hermitian di¬erential operator?

Consider the di¬erential operator

2

D(T ) = {y, T y ∈ L2 [0, 1] : y(0) = y(1) = 0}.

T = ’‚x , (4.36)

With the inner product

1

—

y 1 , y2 = y1 y2 dx (4.37)

0

we have

—

y1, T y2 ’ T y1 , y2 = [y1 y2 ’ y1 y2 ]1 = 0.

—

(4.38)

0

The integrated-out part is zero because both y1 and y2 satisfy the boundary

conditions. We see that

y 1 , T y2 = T y 1 , y2 (4.39)

and so T is Hermitian or symmetric.

The eigenfunctions and eigenvalues of T are

yn (x) = sin nπx

n = 1, 2, . . . . (4.40)

» n = n2 π 2

We see that:

i) the eigenvalues are real ;

variables so that

d ’z— z —

= ’z — e’z z ,

e

dz

—

and that [P (z)] is a function of z — only, so that

d —

[P (z)] = 0.

dz

If you are uneasy at regarding z, z — as independent, you may con¬rm these formulae by

expressing z and z — in terms of x and y, and writing

d 1 ‚ ‚ d 1 ‚ ‚

≡ ’i ≡

, +i .

dz 2 ‚x ‚y dz 2 ‚x ‚y

—

4.2. THE ADJOINT OPERATOR 91

ii) the eigenfunctions for di¬erent »n are orthogonal ,

1

2 sin nπx sin mπx dx = δnm , n = 1, 2, . . . (4.41)

0

√

iii) the normalized eigenfunctions •n (x) = 2 sin nπx are complete: any

function in L2 [0, 1] has an (L2 ) convergent expansion as

√

∞

y(x) = an 2 sin nπx (4.42)

n=1

where

√

1

an = y(x) 2 sin nπx dx. (4.43)

0

This all looks very good ” exactly the properties we expect for ¬nite Her-

mitian matrices! Can we carry over all the results of ¬nite matrix theory to

these Hermitian operators? The answer sadly is no! Here is a counterexam-

ple:

Let

D(T ) = {y, T y ∈ L2 [0, 1] : y(0) = y(1) = 0}.

T = ’i‚x , (4.44)

Again

1

dx {y1 (’i‚x y2 ) ’ (’i‚x y1 )— y2 }