1 — — 1

L† G† (x, y) u(x) ’ G† (x, y) Lx u(x) = Q G† , u

dx . (5.69)

x 0

0

In all cases, the left hand side is equal to

1

dx δ(x ’ y)u(x) ’ GT (x, y)f (x) , (5.70)

0

where T denotes transpose, GT (x, y) = G(y, x). The left hand side is there-

fore equal to

1

u(y) ’ dx G(y, x)f (x). (5.71)

0

5.4. EIGENFUNCTION EXPANSIONS 129

The right hand side depends on the details of the problem. In the present

case, the integrated out part is

1 1

Q(G† , u) = ’ GT (x, y)u(x) = u(0). (5.72)

0 0

At the last step we have used the speci¬c form GT = θ(y ’ x) to ¬nd that

only the lower limit contributes. The end result is therefore the expected

one: y

u(y) = u(0) + f (x) dx. (5.73)

0

It should be clear that variations of this strategy enable us to solve any

inhomogeneous boundary-value problem in terms of the Green function for

the corresponding homogeneous boundary-value problem.

5.4 Eigenfunction Expansions

Self-adjoint operators possess a complete set of eigenfunctions, and we can

expand the Green function in terms of these. Let

L•n = »n •n . (5.74)

Let us further suppose that none of the »n are zero. Then the Green function

has the eigenfunction expansion

•n (x)•— (x )

n

G(x, x ) = . (5.75)

»n

n

That this is so follows from

Lx •n (x) •— (x )

•n (x)•— (x ) n

n

Lx =

»n »n

n n

»n •n (x)•— (x )

n

=

»n

n

•n (x)•— (x )

= n

n

= δ(x ’ x ). (5.76)

Example: : Consider our familiar exemplar

2

D(L) = {y, Ly ∈ L2 [0, 1] : y(0) = y(1) = 0},

L = ’‚x , (5.77)

130 CHAPTER 5. GREEN FUNCTIONS

for which

x(1 ’ x ), x < x ,

G(x, x ) = (5.78)

x (1 ’ x), x > x .

Performing the Fourier series shows that

∞

2

G(x, x ) = sin(nπx) sin(nπx ). (5.79)

n2 π 2

n=1

Modi¬ed Green function

If one or more of the eigenvalues is zero then the modi¬ed Green function is

obtained by simply omitting the corresponding terms from the series.

•n (x)•— (x )

n

Gmod (x, x ) = . (5.80)

»n

»n =0

Then

•n (x)•— (x ).

Lx Gmod (x, x ) = δ(x ’ x ) ’ (5.81)

n

»n =0

We see that this Gmod is still hermitian, and, as a function of x, is orthogonal

to the zero modes. These are the properties we elected in our earlier example.

5.5 Analytic Properties of Green Functions

In this section we will study some of the properties of Green functions con-

sidered as functions of a complex variable. Some of the formul¦ are slightly

easier to derive using contour integral methods, but these are not necessary

and we will not use them here. The only complex-variable prerequisite is a

familiarity with complex arithmetic and, in particular, knowledge of how to

take the logarithm and the square root of a complex number.

5.5.1 Causality Implies Analyticity

If we have a causal Green function of the form G(t ’ „ ) with the property

G(t ’ „ ) = 0, for t < „ , then if the integral de¬ning its Fourier transform,

∞

˜ eiωt G(t) dt,

G(ω) = (5.82)

0

5.5. ANALYTIC PROPERTIES OF GREEN FUNCTIONS 131

converges for real ω, it will converge even better when ω has a positive

˜

imaginary part. This means that G(ω) will be a well-behaved function of

the complex variable ω everywhere in the upper half of the complex plane.

Indeed it is analytic there, meaning that its Taylor series expansion about any

point actually converges to the function. For example, the Green function

for the damped oscillator

1 ’γt

e sin(„¦t), t > 0,

G(t) = (5.83)

„¦

0, t < 0,

has Fourier transform

1

˜

G(ω) = , (5.84)

„¦2 ’ (ω + iγ)2

which is always ¬nite in the upper half-plane, although it has pole singulari-

ties at ω = ’iγ ± „¦ in the lower half-plane.

˜

The only way that the Fourier transform G of a causal Green function

can have a singularity in the upper half-plane is if G contains a exponential

factor growing in time, in which case the system is unstable to perturbations.

This observation is at the heart of the Nyquist criterion for the stability of

linear electronic devices.

Inverting the Fourier transform, we have

1 1 dω