494
G. Kamath
in the general solution to the homogeneous version of (7). Eq. (13) has the general
solution
R(r) =
1
ν
√
r
C 1 J 1
2ν
√
r
+ C 2 Y 1
2ν
√
r
(15)
with J 1 and Y 1 being the Bessel functions of the first and second kind, respectively,
of order 1; and the general solution to the homogeneous version of Eq. (7) will,
therefore, be
φ (r, θ, z) =
∞
−∞
dp
∞
m=0
1
ν
√
r
C 1 J 1
2ν
√
r
+ C 2 Y 1
2ν
√
r
(a m cos m θ + b m sin m θ)
c p θ(p)e −pz + d p θ (−p) e pz
(16)
While Eqs. (12) and (16) are the general solutions, one also has the radial
functions of φ(r) = ln r and φ(r) =
1
r associated with the Coulomb potential
as obvious solutions to Eqs. (4) and (7), respectively; in this context, there have
been several calculations [12–14] on the correction to the hydrogen spectrum from
the inclusion of the Schwarzschild metric, for example, as a perturbation. However,
a complementary investigation involving the quantum mechanics of a hydrogen ion,
for instance, subject to the Coulomb-like potential expressed by (12) and (16) is
clearly preempted as the solutions are infinite series rather than a sum; to this end it
clearly pays to obtain the required answer from the method of Moses [5] tempered
with the orientation due to Plebanski [8] that will be introduced in the latter part
of this paper, if only because it has none of the infirmities that are evident in this
section.
2.1 A Third Example
In 2 + 1 dimensions the metric of Banados, Teitelboim and Zanelli [7] is got from
ds 2 = −
N ⊥ 2 dt 2 +
1
f 2 dr 2 + r 2
dθ + N θ dt
2 ,
2r 2 N θ = −J,
N ⊥ 2 = f 2 = −M +
r 2
l 2 +
J 2
4r 2
(17)
and in polar coordinates it works to
g 00 = M −
r 2
l 2 , g 11 =
1
f 2 , g 02 = −
J
2
, g 22 = r
2
(18)