The Green’s function of the Schr¨odinger equation with the Coulomb potential in the space of constant negative curvature is constructed in the form of an eigenfunction expansion. The closed form expression for this function is also presented

Kurochkin, Yu.

Otchik, V.

BSU Digital Library

Nonlinear Phenomena in Complex Systems, vol. 17, no. 4 (2014), pp. 381 - 383
Coulomb Green’s Function in Lobachevsky Space
Yu. Kurochkin∗ and V. Otchik
B. I. Stepanov Institute of Physics of NAS of Belarus,
68 Nezavisimosti Ave., 220072 Minsk, BELARUS
(Received 05 September, 2014)
The Green’s function of the Schro¨dinger equation with the Coulomb potential in the space
of constant negative curvature is constructed in the form of an eigenfunction expansion. The
closed form expression for this function is also presented.
PACS numbers: 02.40.Ky. 03.65. Nk
Keywords: Green’s function, Lobachevsky space, Schro¨dinger equation, Kepler–Coulomb problem
The quantum mechanical Kepler–Coulomb
problem on a three-dimensional sphere was
ﬁrst solved by Schro¨dinger [1]. Since then,
this problem in spaces of constant curvature
was analyzed by many authors from diﬀerent
viewpoints (see e.g. [2–7] and references therein).
Here we consider Green’s function for the
Schro¨dinger equation with the Coulomb potential
in the three-dimensional space of constant
negative curvature (Lobachevsky space).
We use the embedding of the Lobachevsky
space in a four-dimensional pseudoeuclidean
space with Cartesian coordinates xµ, µ =
1, 2, 3, 4:
xµxµ = x
2 + x24 = x
2 − x20 = −ρ2,
x = {x1, x2, x3}, x4 = ix0.
Commonly used spherical coordinates of the
Lobachevsky space β, θ, φ are deﬁned by relations
x0 = ρ coshβ, x1 = ρ sinhβ sin θ cosφ,
x2 = ρ sinhβ sin θ sinφ,
x3 = ρ sinhβ cos θ,
0 ≤ β <∞, 0 ≤ θ ≤ pi, 0 ≤ φ < 2pi.
(1)
Line and volume elements in these
coordinates are
ds2 = ρ2(dβ2 + sinh2 β(dθ2 + sin2 θdφ2)),
dV = ρ3 sinh2 β sin θdβdθdφ.
(2)
∗E-mail: yukuroch@dragon.bas-net.by
Schro¨dinger equation for a particle in an
external ﬁeld with potential U in the Lobachevsky
space is written in the form (~ = m = 1)
△Ψ+ 2(E − U)Ψ = 0 (3)
where △ is the Laplace operator in the
Lobachevsky space,
△ = − 1
2ρ2
MµνMµν ,Mµν = xµ∂ν − xν∂µ.
Let
1
xµ,
2
xµ denote the Cartesian coordinates
of two diﬀerent points in the Lobachevsky space.
The spherical coordinates of these points are
denoted βi, θi, φi, i = 1, 2. Then the Green’s
function satisﬁes an inhomogeneous equation
[△+ 2(E − U)]G(2x, 1x,E) = δ3(2x, 1x) (4)
where the delta function is deﬁned by relations
δ3(x, x′) =
δ(β − β′)δ(θ − θ′)δ(φ− φ′)
ρ3 sinh2 β sin θ
,∫
δ3(x, x′)f(x′)dV = f(x).
Solutions of equation (4) can be written in the
form of eigenfunction expansions in terms of
simultaneous eigenfunctions of the Hamiltonian
and angular momentum operator.
The Coulomb potential in the Lobachevsky
space we write as
U = −α
ρ
x0
|x| = −
α cothβ
ρ
(5)
381
382 Yu. Kurochkin and V. Otchik
where α > 0 for an attractive potential. The
normalized solutions of the Schro¨dinger equation
(3) with the potential (5) for the continuous and
discrete energy values can be written in the form
ψElm(x) = NES
1
E(β)Y
m
l (θ, φ),
ψnlm(x) = NnSn(β)Y
m
l (θ, φ)
(6)
where Y ml (θ, φ) are the spherical harmonics,
and dependence on β is expressed in terms of
hypergeometric functions:
S1E(β) = sinh
l βe(ip−l−1)β2F1(σ + l + 1, l − ν + 1; 2l + 2; 1− e−2β)
(p =
√
2Eρ2 + 2αρ− 1, ν − σ = ip, σ = αρ/ν);
Sn(β) = sinh
l βe(−αρ/n+n−l−1)β2F1(αρ/n+ l + 1, l − n+ 1; 2l + 2; 1− e−2β).
The normalization coeﬃcients are
NE =
2l+1/2
√
sinhpip|Γ(l + ν + 1)||Γ(l + σ + 1)|
pi
√
ρ(2l + 1)!
,
Nn =
2l+1
(2l + 1)!
√(
α2
ρn3
− n
ρ3
)
(n+ l)!Γ(αρ/n+ l + 1)
(n− l − 1)!Γ(αρ/n− l) .
The eigenfunction expansion for the
Coulomb Green’s function in terms of the
solutions (6) is
G(
2
x,
1
x,E0) =
∞∑
l=0
l∑
m=−l
∞∫
−
α
ρ
+ 1
2ρ2
ψElm(
2
x)ψ∗Elm(
1
x)
E0 − E dE
+
[αρ]∑
n=1
n−1∑
l=0
l∑
m=−l
ψnlm(
2
x)ψ∗nlm(
1
x)
E0 − En .
(7)
In the integral (7), E0 is taken on the upper
side of the real axis (ImE0 > 0) to provide the
diverging wave behaviour of the Green’s function
(see discussion in [8] for the case of a ﬂat space).
By performing integration in E, we arrive at
the characteristic form of Green’s function
G(
2
x,
1
x,E) = − 1
4pi
∞∑
l=0
Γ(l + 1− ν)Γ(l + σ + 1)S1E(β<)S2E(β>)Pl(cos θ12)
ρΓ(1− ip)(2l)! (8)
where S2E(β) = sinh
−l−1 βe(ip+l)β2F1(σ − l,−ν − l; 1− ip; e−2β),
Нелинейные явления в сложных системах Т. 17, № 4, 2014
Coulomb Green’s Function in Lobachevsky Space 383
β> =
{
β1, if β1 > β2,
β2, if β2 > β1,
and similar for β< , cos θ12 = cos θ1 cos θ2 + sin θ1 sin θ2 cos(φ1 − φ2).
In papers of Hostler [9, 10], a closed
form expression for the nonrelativistic Coulomb
Green’s function in a ﬂat space has been obtained
by summing the partial wave expansion. It is
possible to extend Hostler’s approach to the case
of the Lobachevsky space. We only formulate here
a ﬁnal result. By using notations
f1E(β) = sinhβS
1
E0(β) = sinhβe
(ip−1)β
2F1(σ + 1, 1− ν; 2; 1− e−2β),
f2E(β) = sinhβS
2
E0(β) = e
ipβ
2F1(σ,−ν; 1− ip; e−2β).
we can write a closed expression for the Coulomb Green’s function in the Lobachevsky space:
G(
2
x,
1
x,E) = − Γ(1− ν)Γ(σ + 1)
4piρ sinhβ12Γ(1− ip)
(
∂
∂β−
− ∂
∂β+
)
f1E(β−)f
2
E(β+) (9)
where
β± = (β1 + β2 ± β12)/2, coshβ12
= coshβ1 coshβ2 − sinhβ1 sinhβ2 cos θ12.
References
[1] E. Schro¨dinger. Proc. R. Irish. Acad. A. 46, 9
(1940); 46, 183 (1941); 47, 53 (1941).
[2] L. Infeld, A. Schild. Phys. Rev. 67, 121 (1945).
[3] P.W. Higgs. J. Phys. A: Math. and Gen. 12, 309
(1979).
[4] H.I. Leemon. J. Phys. A: Math. and Gen. 12, 489
(1979).
[5] A.O. Barut, A. Inomata, G. Junker. J. Phys. A:
Math. and Gen. 20, 6271 (1987).
[6] Ya.A. Granovsky, A.S. Zhedanov, I.M. Lutsenko.
Teor. Mat. Fiz. 91, 396 (1992).
[7] C. Grosche, G.S. Pogosyan, A.N. Sissakian. Part.
Nucl. 28, 486 (1997).
[8] A.I. Baz’, Ya.B. Zel’dovich, A.M. Perelomov.
Rasseyanie, reaktsii i raspady v nerelativistskoi
kvantovoi mekhanike. (Nauka, Moscow, 1971).
(In Russian).
[9] L. Hostler, R.H. Pratt. Phys. Rev. Lett. 10, 469
(1963).
[10] L. Hostler. Journ. Math. Phys. 5, 591 (1964).
Nonlinear Phenomena in Complex Systems Vol. 17, no. 4, 2014


Coulomb Green’s Function in Lobachevsky Space

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Coulomb Green’s Function in Lobachevsky Space

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