QuantumPhysics.dvi

To solve (12.3) perturbatively, we move the perturbative correction piece to the right hand side
of the equation,

(H 0 −E)|ψk〉=−λH 1 |ψk〉 (12.4)

The operatorH 0 −Eis not quite invertible, and precisely has the states|φk〉for its null-space. In
the case of bound state perturbation theory, we remedied this situation byinvertingH 0 −Eon
the subspace of the Hilbert space orthogonal to the its null-space. This was possible because the
null-space consisted of discrete states. In the case of the continuous spectrum, the natural way of
dealing with the null-space is as follows. We add a small imaginary partiεto the energyE. Since
H 0 is self-adjoint, all its eigenvalues are real, and thus the modified operatorH 0 −E±iεwill now
be invertible, although the eigenvalue corresponding to states|φk〉in the null-space will be tiny.
This procedure naturally produces two solutions to (12.3) and (12.4), given by

|ψ±k〉=|φk〉−

λ

H 0 −E∓iε

H 1 |ψk±〉 ε > 0 (12.5)

This equation is known as theLippmann-Schwinger equation. (Later on, we shall use an even more
drastic extension of this equation and letz=E±iεtake arbitrary values in the complex plane.)
Althoughεwas introduced here as a finite real positive number, it is understood that one must
take the limitε→0. To understand thisiεprescription in more detail, it will be helpful to study
its mathematical properties in some detail.

12.2 Theiεprescription

Theiεprescription is given by the limit

lim

ε→ 0

1

x−iε

x∈R, ε > 0 (12.6)

should be understood as a generalized function (ordistribution) similar to the Diracδ-function. In
fact, it contains aδ-function, as may be seen by separating its real and imaginary parts,

1

x−iε

=

x

x^2 +ε^2

+

iε

x^2 +ε^2

(12.7)

The limit of the first term is, by definition, the sl principal value prescription PV(1/x), while the
limit of the second may be found by integrating it against a smooth test functionf(x),

lim

ε→ 0

∫

R

dx

iε

x^2 +ε^2

f(x) = lim

ε→ 0

∫

R

dx

i

x^2 + 1

f(εx) =iπf(0) (12.8)

Hence the limit of the second term is given byiπδ(x). Putting all together, we obtain,

lim

ε→ 0

1

x−iε

= PV

1

x

+iπδ(x) (12.9)

This clearly demonstrates that the limit is a generalized function. It is customary when writing
theiεprescription to simply omit the limε→ 0 symbol in front.