real and imaginary parts:

n(ω) = nR + inI

i

= nR (ω) + γ(ω), (5.114)

2k

where γ is the extinction coe¬cient, de¬ned so that the intensity falls o¬ as

I = I0 exp(’γx). A non-zero γ can arise from either energy absorbtion or

scattering out of the forward direction2 . For the refractive index, we have

the Kramers-Kronig relation

c γ(ω )

∞

nR (ω) = 1 + P dω . (5.115)

ω 2 ’ ω2

π 0

Formul¦ like this will be rigorously derived later by the use of contour-

integral methods.

5.5.3 Resolvent Operator

Given a di¬erential operator L, we de¬ne the resolvent operator to be R» ≡

(L ’ »I)’1 . The resolvent is an analytic function of », except when » lies in

the spectrum of L.

We expand R» in terms of the eigenfunctions as

•n (x)•— (x )

n

R» (x, x ) = . (5.116)

»n ’ »

n

When the spectrum is discrete, the resolvent has poles at the eigenvalues

L. When the operator L has a continuous spectrum, the sum becomes an

integral:

•µ (x)•— (x )

µ

R» (x, x ) = ρ(µ) dµ, (5.117)

µ’»

µ∈σ(L)

2

For a dilute medium of incoherent scatterers, such as the air molecules resposible for

Rayleigh scattering, γ = N σ tot , where N is the density of scatterers and σ tot is the total

scattering cross section of each.

138 CHAPTER 5. GREEN FUNCTIONS

where ρ(µ) is the eigenvalue density of states. This is of the form that

we saw in connection with the Plemelj formul¦. Consequently, when the

spectrum comprises segements of the real axis, the resulting analytic function

R» will be discontinuous across the real axis within them. The endpoints

of the segements will branch point singularities of R» , and the segements

themselves, considered as subsets of the complex plane, are the branch cuts.

The trace of the resolvent Tr R» is de¬ned by

dx {R» (x, x)}

Tr R» =

•n (x)•— (x)

n

= dx

»n ’ »

n

1

=

n »n ’ »

ρ(µ)

’ dµ. (5.118)

µ’»

Applying Plemelj to R» , we have

Im lim Tr R»+iµ = πρ(»). (5.119)

µ’0

Here, we have used that fact that ρ is real, so

—

Tr R»’iµ = Tr R»+iµ . (5.120)

The non-zero imaginary part therefore shows that R» is discontinuous across

the real axis at points lying in the continuous spectrum.

Example: Consider

L = ’‚x + m2 ,

2

D(L) = {y, Ly ∈ L2 [’∞, ∞]}. (5.121)

As we know, this operator has a continuous spectrum, with eigenfunctions

1

•k = √ eikx . (5.122)

L

Here, L is the (very large) length of the interval. The eigenvalues are E =

k 2 + m2 , so the spectrum is all positive numbers greater than m2 . The

momentum density of states is

L

ρ(k) = . (5.123)

2π

5.5. ANALYTIC PROPERTIES OF GREEN FUNCTIONS 139

The completeness relation is

dk ik(x’x )

∞

= δ(x ’ x ),

e (5.124)

2π

’∞

which is just the Fourier integral formula for the delta function.

The Green function for L is

•k (x)•— (y) dk eik(x’y)

dn 1 ’m|x’y|

∞ ∞

k

G(x ’ y) = dk = = e .

k 2 + m2 ’∞ 2π k 2 + m2

dk 2m

’∞

(5.125)

We can use the same calculation to look at the resolvent R» = (’‚x ’ »)’1 .

2

Replacing m2 by ’», we have

√

1

√ e’ ’»|x’y| .

R» (x, y) = (5.126)

2 ’»

√

To appreciate this expression, we need to know how to evaluate z where

z is complex. We write z = |z|eiφ where we require ’π < φ < π. We now

de¬ne √

z = |z|eiφ/2 . (5.127)

√

When we evaluate z for z just below the negative real axis then this de¬ni-

tion gives ’i |z|, and just above the axis we ¬nd +i |z|. The discontinuity

means that the negative real axis is a branch cut for the the square-root

√

function. The ’»™s appearing in R» therefore mean that the positive real

axis will be a branch cut for R» . This branch cut therefore coincides with

the spectrum of L, as promised earlier.

Im »

»

Re »

’» arg( ’»)/2

’»

√

If Im » > 0, and with the branch cut for z in its usual place along the

√

negative real axis, then ’» has negative imaginary part and positive real

part.

140 CHAPTER 5. GREEN FUNCTIONS

If » is positive and we shift » ’ » + iµ then