16
2 The Discrete Spectrum and the Continuum
energies. As it possesses bound, resonance, and scattering eigenstates, contrary to
the harmonic oscillator potential, it is of great importance for the study of continuum
degrees of freedom in nuclei.
In this chapter, one will also deal with analytic properties of the eigenfunctions
of the radial Schrödinger equation with respect to their linear momentum, denoted
k, in the complex momentum plane. Indeed, analyticity of eigenfunctions in k is
necessary to demonstrate that bound and scattering states form a complete set of
states (see Sect. 3). One will also introduce resonance states which are eigenstates of
complex energy embedded in the continuum of scattering states. As they physically
correspond to long-lived states, they are of fundamental importance for the study of
weakly bound and unbound nuclei.
2.1
Definition of One-Body States
For one-body Hamiltonians bearing spherical symmetry, the solution of the
Schrödinger equation separates into two parts, a spherical harmonics and a radial
wave function [2]:
φ(k, r) =
u(k, r)
r
Y m (θ, ϕ) .
(2.1)
The properties of the spherical harmonics Y
m (θ, ϕ) are well known and described
for example in Ref. [2]. The radial wave function u(k, r) obeys the Schrödinger
equation:
u
(k, r) =
+ 1)
r 2
+ v l (r) − k
2
u(k, r) ,
(2.2)
where v l (r) is the potential, is the (positive) orbital angular momentum, and k is
the linear momentum which can be complex. Potentials will be considered to be
real on the real axis unless stated otherwise. The Hamiltonian of Eq. (2.2) can be
formally written as an operator:
h = −
d 2
dr 2 +
+ 1)
r 2
+ ˆ
v l .
(2.3)
Equation (2.2) bear the limiting cases for r → 0 and r → +∞:
u
(k, r) =
0 (( 0 + 1)
r 2
+
v c 0
r
+ v 0 − k
2
u(k, r) , r → 0
(2.4)
u
(k, r) =
+ 1)
r 2
+
v c
r
− k
2
u(k, r) , r → +∞
(2.5)
2 The Discrete Spectrum and the Continuum
energies. As it possesses bound, resonance, and scattering eigenstates, contrary to
the harmonic oscillator potential, it is of great importance for the study of continuum
degrees of freedom in nuclei.
In this chapter, one will also deal with analytic properties of the eigenfunctions
of the radial Schrödinger equation with respect to their linear momentum, denoted
k, in the complex momentum plane. Indeed, analyticity of eigenfunctions in k is
necessary to demonstrate that bound and scattering states form a complete set of
states (see Sect. 3). One will also introduce resonance states which are eigenstates of
complex energy embedded in the continuum of scattering states. As they physically
correspond to long-lived states, they are of fundamental importance for the study of
weakly bound and unbound nuclei.
2.1
Definition of One-Body States
For one-body Hamiltonians bearing spherical symmetry, the solution of the
Schrödinger equation separates into two parts, a spherical harmonics and a radial
wave function [2]:
φ(k, r) =
u(k, r)
r
Y m (θ, ϕ) .
(2.1)
The properties of the spherical harmonics Y
m (θ, ϕ) are well known and described
for example in Ref. [2]. The radial wave function u(k, r) obeys the Schrödinger
equation:
u
(k, r) =
+ 1)
r 2
+ v l (r) − k
2
u(k, r) ,
(2.2)
where v l (r) is the potential, is the (positive) orbital angular momentum, and k is
the linear momentum which can be complex. Potentials will be considered to be
real on the real axis unless stated otherwise. The Hamiltonian of Eq. (2.2) can be
formally written as an operator:
h = −
d 2
dr 2 +
+ 1)
r 2
+ ˆ
v l .
(2.3)
Equation (2.2) bear the limiting cases for r → 0 and r → +∞:
u
(k, r) =
0 (( 0 + 1)
r 2
+
v c 0
r
+ v 0 − k
2
u(k, r) , r → 0
(2.4)
u
(k, r) =
+ 1)
r 2
+
v c
r
− k
2
u(k, r) , r → +∞
(2.5)