2.6 Analytical Properties of the Wave Functions
59
is defined from Jost functions and hence C ± constants:
S (k) =
J − (k)
J + (k)
= −
C +
C − .
(2.182)
The fact that the potential v l (r) is real in Eq. (2.2) implies that C + = C − ∗ if k > 0
(see Eq. (2.7)). One cannot have C ± = 0 for k > 0, as otherwise u(k, r) = 0
∀r ≥ 0. Thus, |S (k)| = 1 on the real k-axis.
This relation is important physically as it implies the flux conservation in a
reaction process. As a consequence, using Eq. (2.169), it is possible to define S (k)
on the real k-axis as:
S (k) = exp(2iδ (k)) ,
(2.183)
where δ (k) is called the phase shift of the scattering state u(k, r). In particular,
it varies very quickly close to a narrow resonance, so that one can identify longlived states in the positive energy region. The bound states and narrow resonances
of linear momentum k are associated to the poles of the S-matrix, because S(k) is
infinite therein, as can be seen from Eq. (2.182). The residues of the poles of the
S-matrix are calculated in Exercise XIV.
Exercise XIV
Calculate the residues of S (k) at its nonzero poles k n using Eq. (2.174).
Due to Eq. (2.169), the S-matrix is analytical over the complex plane, except for
its poles and a cut along the real negative k-axis in the case of a Coulomb potential.
In fact, the analyticity of S-matrix, that is, the fact the S-matrix is a function of k
only, is a prerequisite for it to be able to describe physical observables. Indeed, as
the state having k ∗ as linear momentum is the time reversed state of a resonance
bearing k as linear momentum, a function of both k and k ∗ would be influenced by
both past and future, thus violating the causality principle.
The nature of the poles of the S-matrix is of importance, as they correspond
to bound states or resonances. One has already demonstrated in the previous
section that bound states can exist only if k is purely imaginary (see Exercise XI).
Consequently, the only poles that the S-matrix can have in the upper complex plane
correspond to real-energy bound states and lie on the positive imaginary-k axis.
There is no such requirement in the lower half-plane so that complex poles can exist
therein, and they indeed correspond to unbound states.
In the first place, one will consider (k) ≥ 0. The S-matrix poles of physical
importance are those close to the real k-axis as they represent the resonance states
bearing a long lifetime. In the interval −π/4 < arg(k) < 0, real and imaginary parts
of k and k 2 have the same sign, so in the expression: k 2 = E − i
Γ
2 , both E and Γ
are positive. The physical significance of E and Γ becomes explicit when put in the
59
is defined from Jost functions and hence C ± constants:
S (k) =
J − (k)
J + (k)
= −
C +
C − .
(2.182)
The fact that the potential v l (r) is real in Eq. (2.2) implies that C + = C − ∗ if k > 0
(see Eq. (2.7)). One cannot have C ± = 0 for k > 0, as otherwise u(k, r) = 0
∀r ≥ 0. Thus, |S (k)| = 1 on the real k-axis.
This relation is important physically as it implies the flux conservation in a
reaction process. As a consequence, using Eq. (2.169), it is possible to define S (k)
on the real k-axis as:
S (k) = exp(2iδ (k)) ,
(2.183)
where δ (k) is called the phase shift of the scattering state u(k, r). In particular,
it varies very quickly close to a narrow resonance, so that one can identify longlived states in the positive energy region. The bound states and narrow resonances
of linear momentum k are associated to the poles of the S-matrix, because S(k) is
infinite therein, as can be seen from Eq. (2.182). The residues of the poles of the
S-matrix are calculated in Exercise XIV.
Exercise XIV
Calculate the residues of S (k) at its nonzero poles k n using Eq. (2.174).
Due to Eq. (2.169), the S-matrix is analytical over the complex plane, except for
its poles and a cut along the real negative k-axis in the case of a Coulomb potential.
In fact, the analyticity of S-matrix, that is, the fact the S-matrix is a function of k
only, is a prerequisite for it to be able to describe physical observables. Indeed, as
the state having k ∗ as linear momentum is the time reversed state of a resonance
bearing k as linear momentum, a function of both k and k ∗ would be influenced by
both past and future, thus violating the causality principle.
The nature of the poles of the S-matrix is of importance, as they correspond
to bound states or resonances. One has already demonstrated in the previous
section that bound states can exist only if k is purely imaginary (see Exercise XI).
Consequently, the only poles that the S-matrix can have in the upper complex plane
correspond to real-energy bound states and lie on the positive imaginary-k axis.
There is no such requirement in the lower half-plane so that complex poles can exist
therein, and they indeed correspond to unbound states.
In the first place, one will consider (k) ≥ 0. The S-matrix poles of physical
importance are those close to the real k-axis as they represent the resonance states
bearing a long lifetime. In the interval −π/4 < arg(k) < 0, real and imaginary parts
of k and k 2 have the same sign, so in the expression: k 2 = E − i
Γ
2 , both E and Γ
are positive. The physical significance of E and Γ becomes explicit when put in the