We report on our analysis of the Coulomb singularity problem in the frame of the coupled channel scattering theory including spin-orbit interaction. We assume that the coupling between the partial wave components involves orbital angular momenta such that Δl = 0, ±2. In these conditions, the two radial functions, components of a partial wave associated to two values of the angular momentum l, satisfy a system of two second-order ordinary differential equations. We examine the difficulties arising in the analysis of the behavior of the regular solutions near the origin because of this coupling. First, we demonstrate that for a singularity of the first kind in the potential, one of the solutions is not amenable to a power series expansion. The use of the Lippmann-Schwinger equations confirms this fact: a logarithmic divergence arises at the second iteration. To overcome this difficulty, we introduce two auxilliary functions which, together with the two radial functions, satisfy a system of four first-order differential equations. The reduction of the order of the differential system enables us to use a matrix-based approach, which generalizes the standard Frobenius method. We illustrate our analysis with numerical calculations of coupled scattering wave functions in a solid-state system.

1.
H. I.
Ralph
,
Philips Res. Rep.
32
,
160
(
1977
).
2.
F.
Calogero
,
Variable Phase Approach to Potential Scattering
(
Academic
,
New York
,
1967
).
3.
P.
Bogdanski
and
H.
Ouerdane
,
Phys. Rev. B
74
,
085210
(
2006
).
4.
R. G.
Newton
,
Scattering Theory of Waves and Particles
(
McGraw-Hill
,
New York
,
1966
).
5.
J. R.
Taylor
,
Scattering Theory, the Quantum Theory of Nonrelativistic Collisions
(
Dover
,
New York
,
2009
).
6.
M.
Reed
and
B.
Simon
,
Scattering Theory, Volume III of Methods of Modern Mathematical Physics
(
Academic
,
New York
,
1979
).
7.
J. R.
Meyer
and
F. J.
Bartoli
,
Phys. Rev. B
23
,
5413
(
1981
).
8.
A.
Baldereschi
and
N. O.
Lipari
,
Phys. Rev. B
8
,
2697
(
1973
).
9.
A.
Ronveaux
,
Am. J. Phys.
37
, (
1969
).
10.
R. G.
Newton
,
J. Math. Phys.
1
,
319
(
1960
).
11.
J. R.
Cox
and
A.
Perlmutter
,
Nuovo Cimento
37
,
76
(
1965
).
12.
E. L.
Ince
,
Ordinary Differential Equations
(
Dover
,
New York
,
1926
).
13.
E. A.
Coddington
and
N.
Levinson
,
Theory of Ordinary Differential Equations
(
McGraw-Hill
,
New York
,
1955
).
14.
A potential of the form v(x) = V0exp ( − ax)/x, for which all the terms in the expansion are non-zero.
15.
v(x) = V1 for x < x0 and v(x) = 0 elsewhere.
16.
M.
Lannoo
and
G.
Allan
,
Solid State Commun.
33
,
293
(
1980
).
17.
F.
John
,
Ordinary Differential Equations
(
Courant Institute of Mathematical Sciences
,
New York
,
1965
).
18.
J.
Joachain
,
Quantum Collision Theory
(
North-Holland
,
Amsterdam
,
1983
).
19.
W.
Greub
,
Linear Algebra
(
Springer-Verlag
,
Berlin
,
1981
).
20.
P.
Bogdanski
, D.Sc. dissertation,
Université de Caen
,
2003
.
21.
C. H.
Edwards
and
D. E.
Penney
,
Elementary Differential Equations and Applications
(
Prentice Hall
,
Englewood Cliffs, NJ
,
1989
).
22.
N. O.
Lipari
and
A.
Baldereschi
,
Phys. Rev. Lett.
25
,
1660
(
1970
).
You do not currently have access to this content.