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The path integral for the Kepler problem on the pseudosphere
ANNALS
OF PHYSICS
204, 208-222
(1990)
The Path Integral for the Kepler Problem on the Pseudosphere CHRISTIAN
Technology
The Blackett and Medicine,
GROSCHE*
Laboratory, Imperial College of Science, Prince Consort Road, London SW7. United Received
February
Kingdom
20, 1990
The path integral for the generalized Kepler problem on the pseudosphere is calculated. As the generalized Kepler problem we describe the potential problem P’(a) = - (Ze*/R)( coth G(- 1) (a >O, R-curvature) on the D-dimensional pseudosphere, whereas for the genuine Kepler problem this feature of the potential is only true for the four-dimensional pseudosphere. Energy spectrum and wave-functions are explicitly calculated. The result is compared with the flat-space limit (i.e., R + “c) and with the free motion on the pseudosphere. The application of the result of the path integral solution of the Manning-Rosen potential plays a crucial role in the calculation. f’ 1990 Academic Press, Inc.
I. INTRODUCTION In this paper I study a specific potential problem on the D-dimensional pseudosphere, the “generalized Kepler problem.” The pseudosphere is a space of constant negative curvature. Spaceslike this have recently become important in the study of strings and quantum chaos. Whereas in the former it turns out that Laplacians on bounded domains in these spacesare relevant in the partition function 2 of the Polyakov [41] approach to string theory [ 10, 16, 18,371, in the latter one studies the dynamical properties of classical and quantum motion in these spaces [ l-3,26,42]. However, also quantum mechanical considerations on these spaces are important, especially calculations by path integrals, where the emphasis lies in the attempt to build up quantum mechanics on an equal footing, on the one hand, by operator calculus and, on the other, by Feynman’s path integration [14]. In an earlier paper we have already studied the free motion on the pseudosphere [25], including its connection in the three-dimensional case to the Poincare upper half-plane [24] and the Poincare disc [20]. Further contributions are due to Gutzwiller [26] and Bohm and Junker [S]. Also the effect of including a magnetic field was studied [21]. * Supported
by Deutsche
Forschungsgemeinschaft
under
208 0003-4916/90
$7.50
Copyright Ci 1990 by Academic Press, Inc. All rights of reproductmn in any form reserved.
Contract
DFG
Gr 1031
KEPLER PROBLEMON
THE PSEUDOSPHERE
209
The study of the Kepler problem is quite old and was motivated to compare the properties of the Coulomb potential in an “open hyperbolic” universe to that of an “open but flat” universe. As Infeld and Schild [28] found, the main difference is simply the fact that in an open hyperbolic universe there are only a finite (but very large) number of bound states. The same, of course, can be done for a closed universe, a problem first discussedby Schrijdinger [41]. Other attempts to the latter problem are due to Infeld [27], Stevenson [44], and Barut and Wilson 171, whereas the path integral treatment is due to Barut, Inomata, and Junker [S]. The further content of this paper is as follows: In the next section I construct the path integral on the D-dimensional pseudosphere /1” ‘. In Section III the generalized Kepler problem is formulated and its path integral is calculated. Whereas the notion “Kepler problem” is only true in the strict sensefor the D = 4 case, it can be easily generalized to the general D case, because I’ depends only on the variable LX.Due to this property the angular variables can be easily separated, leading eventually to the path integral problem for the Manning-Rosen potential which has already been solved in Ref. [22]. Using this solution, the path integral for the generalized Kepler problem on the pseudosphereis immediately obtained. In the flat space limit (i.e., R + t-x) we can recover the result of the I/r-potential problem in n= D - 1 dimensions for the discrete and continuous state wavefunctions and energy-spectra, respectively. For the case of vanishing potential strength the known results of the free motion on the pseudosphere are obtained. The last section contains some concluding remarks.
II. CONSTRUCTING
THE PATH INTEGRAL
Let us sketch the canonical way to construct a path integral on a curved manifold [ll, 12, 15, 23, 30, 33-36, 381. One considers the generic case, where the classical Lagrangian is given by
(1) with the metric tensor g,, and line elements d.?= guhdqudqh. The quantum Hamiltonian reads
where A,, is the LaplaceeBeltrami operator, g = det( gab) and g”’ = (go,,) ‘. One considers momentum operators
(3)
210
CHRISTIAN
which are hermitian
GROSCHE
with respect to the scalar product (fi,fi)
= j” Ji
dqf3q)fAq).
(4)
The crucial point in the construction of the path integral is the question of which ordering prescription one should use in the Hamiltonian. One well-known ordering rule is the Weyl-ordering which yields for the Hamiltonian
ff=&
(g”b~,pb+2p,g”bpb+~,pbg”b)+
with the well-defined
quantum A Vwey, = &
=r(iii
v(q
I+ A Vweydq
(5)
1
potential ( gabT;J;,-
[fbrorb
R’)
+
2k”brb),,
+
(6)
fb,ubl.
Here R’ denotes the scalar curvature and r;, are the Christoffel symbols. The resulting path integral constructed from the Hamiltonian (5) is then given in the mid-point lattice description:
K(q”, q’; T) = [g(q’)
g(q”)]
x exp
{I
-‘I4 N.-(~)ND’2(~~,‘Idy(j)),~~~ lim
5 2 g,,(q”‘)
Aq”‘“Aq(j’b
- cV(q(j’)
-I>
- &A V(q(j))
,
(7)
where q(j) = q(t’ +jE), g(l) = $($- 1) + qlj)), Aq(A = q(A _ q(j- I), E = T/N, T= f _ tt in the limit N + co. (I use the symbol A for different purposes; however, the relevant meaning should be clear by context.) An alternative approach is a product ordering [ 191 of the quantum Hamiltonian H. Here it is assumed that g,, can be written as g,, = h,,h,,; then ff=$f%,p,hcb+ with the well-defined AV,,
quantum = AVwey, +;
V(q)+AV,,(q)
(8)
potential (2hnchbr.ab - har,ohbr,b - hac,bhbc,a).
(9)
KEPLER
The resulting
PROBLEM
ON
THE
PSEUDOSPHERE
211
path integral reads
K(q”, q’; T) = N-r lim (&)ND~2‘~~,‘j&i??4~~’
-&V(q”‘)-edI/,,(q”‘) In order to specify our problem we must define g,, and V. Whereas P’ will be fixed in the next section, I show now the construction of g,, and the corresponding path integral on the pseudosphere. I introduce D-dimensional pseudospherical polarcoordinates [45] (see also [25] for some more details), x,=Rcoshcc x2 = R cash M cos 0,
2
sz=RcoshcrsinO,
,cosODP,
(11)
s D ,=RcoshccsinODP,...sinHzcos~
xD = R cash tl sin 0,
z . . sin I32sin 4,
where CI2 0, 0 < 8,, d 71(v = 2, .... D - 2) and 0 < 4 < 271(I also use for shorthand the notation 8,-, = a and 0, = 4). The metric tensor in x-space reads G,h= (- 1, 1, .... 1) (a, b = 1, .... D) such that .x?= -R’= -.~f + Cf=‘=, .x: with R fixed defines the D-dimensional pseudosphereAD ‘. The Gaussian curvature reads K= -(Dl)(D-2)/R’. The metric tensor on the pseudosphere is given by g,,= R’(1, sinh’ ~1, sinh’ c1sin’ 8,-,, .... sinh’ cx.. .sin’ 02) (a, b = 1, .... D- 1). Therefore, for the Hamiltonian
+...+
smh’ CX.. sin’ 8? &j
212
CHRISTIAN GROSCHE
with the quantum potential AY(O)=&[(D-2&y--
... -
1 sinh’ CI.. . sin2 e2I
(13)
({ 0) denotes a collection of variables). Due to the specific nature of g,, we have AV Wey,= AV,, E AV. Thus we obtain the (Lagrangian) path integral on the D-dimensional pseudosphere AD ~ I:
SinhD-2
cl(j)
&(j)
&J(J)
s x exp i ,g 1
[Lg,({e(j-
I)}, { e(jJ})-Ev({e(j)})-Edv({e(j)})]
J=I
.
(14)
I
L,, denotes the classical Lagrangian, [oi2+sinh2&iP2+
... +(sinh2cr...sin2e2)$2],
(15)
and its counterpart on the lattice reads
Lg,({e(j-l)), {e(q) n =$[A Here f2(q”‘)
2x(J)
=f(q”-
+
sinh2
E(i)A20(/)
’ ‘f(q(j’)
D-2
+
. .
n A24(j)] . + (sinh2 ~1~‘). . . sin2e(jl) 2
.
(16)
and D-2
dQ = n
(sin ek)k-l de,
(17)
k=l
denotes the (D - 2)-dimensional surface element on the unit-sphere SD-’ [ 13, Chap. XI]. This completes the construction of the path integral on the D-dimensional pseudosphere AD- ‘.
III. THE KEPLER PROBLEM The quantum mechanical Kepler problem in a space of constant negative curvature was first discussedby Infeld and Schild [28] by the factorization method. They introduced it to compare its energy spectrum of a hydrogen atom in an open
KEPLER
hyperbolic universe to that difference lies in the fact that levels is finite, however very sional hyperbolic space with
ds’ =
dr’
1 - r’/R’
+ r’(d0’
PROBLEM
213
ON THE PSEUDOSPHERE
in an open flat universe. We shall find that the main in an open hyperbolic universe the number of discrete large indeed. The line element of the (three-) dimenthe coordinates (r, 8,$) can be written in the form
+ sin’ 8 dq5’)
(~>0,HECO,7Cl,~ECO,27Cl), (1)
= R’[dc?+ sinh’ @do2 + sin’ 8 d@)]
where we have set sinh r = r/R. Note that the dimension of this d= 3 dimensional hyperbolic space corresponds to the D = 4 dimension of the pseudosphere n3. The potential energy V(x) of an electron in a central ‘Coulomb-like” field must satisfy Poisson’s equation A,,
V(m) = -47TZe%(a),
(2)
where d,, is the Laplace-Beltrami operator in the three-dimensional hyperbolic space. Thus we find for V, V(a) = - g if we demand that V+O
(coth x - 1),
as C(+(XCIand V-+
-Ze’/r
as R-+m
(R-tee,
aR=r
fixed).
Although this potential is in the strict senseonly Coulombian for D = d + 1 = 4 we can easily generalize it to arbitrary D or d, respectively. This is, of course, similar to the consideration of the l/r-potential problem in flat d-dimensional space [9], where only for d= 3 do we have the genuine Coloumb/hydrogen problem. In the following I will omit the d and D superscripts, respectively, and denote by the “generalized Kepler problem on the pseudosphere” the potential problem (3) on AD ~ ‘. Thus I obtain the path integral
.
(4)
d V( 10) ), of course, as in Eq. (II.1 3). Due to the very singular nature of d V( (0)) this path integral is at it stands is not tractable. However, we can use a path integral identity (based on a method developed in [ 15, 391) already derived in
214
CHRISTIAN
GROSCHE
Ref. [23] to simplify the path integration significantly and separate the angular variables BoPz, .... 4 from the hyperbolic coordinate CY.I introduce the quantity $(‘.” defined by
D 3 c cosfycosH),~’ rn= I
cos I) “~2’=cosOj:‘2COS8~‘r+
which is actually the addition theorem on the S”- ‘-sphere and cos $“.” = Q(“..Q’“, where Q’i.” are unit vectors on the S” ‘-sphere. Using the result of Ref. [23] the following path integral identity can be achieved (replace R = r”) = sinh cx”’ in Eq. (2.27)): exp
imR’ ci, 2Ehsmh-cr
[d’6)g)z+
1 :+ ... +sin2(j~~2...sin’(j~~
h*(D-2)’ 8mR’
Now expanding
e:cos$w 7 =
according
7 v : z-(v) f. (Ifv) z 0 /=O
[ 11 ]-to denote The highly singular
sinh”P’rDr(f)i’
[ti’ + 2 sinh” a( 1 - cos $)] + g
the exponential
(6)
>I
A Pfollowing Dewitt Here I have used the symbol “equivalence as far as use in the path integral is concerned.” terms are cancelling and I obtain
-
A” #(I)]
id 8mR’ sinh’ %“I
(1 -co~Ic/“~‘.“)-
sln2(jgJ
? . sin&$))
... +(sin/;;;i;:’
DO(r) coth c(
(7)
to [ 17, p. 9801, C;‘(cos
l+v’y
Z,+.(z),
(8)
together with [ 13, Chap. XI]
f Sf’(D”‘)S~(R”~)=2*(D1~l~~2r(~)2f~”33c~D~~~~~(cosiil-‘)), (9) p = 1
KEPLER
PROBLEM
ON
THE
215
PSEUDOSPHERE
where the S?(0) denote the real hyper-spherical harmonics of degree 1 with vector Q on the S” -- ’ -sphere, I~N~and~=l,...,M,M=(21+D-2)(1+D-3)!/ f!(D-2)!. Thus for v=(D-3)/2: e:cos*“~” = 2X 271 ( z> Therefore
(D-3)/2
% A4 c c s~(~“‘)s~(~“‘)z,+,,~~,,,2(=). /=o I’=1
I obtain for the path integral ~(Kep’er)( (o”},
{y};
unit
(10)
(7)
T) h’(D-2)’ 8mR2
!I
The kernel K, is given by
x
p,[sinh’
z] Dcc( t)
>1
a At , with the functional p,[sinh’
(12)
measure ~~[sinh’ E], I\ CX]= lim p,[sinh’ ~“‘1 N-r N
= lim n WV-.%,=I
znrnR’ /‘--A ~ smh’ tl(” i&h
x exp ~sinhC%)) (
“2
I,+,.~-,,,,
(gsinG/)).
(13)
TO get into contact with a one-dimensional potential problem the asymptotic expansion I,,(z) 2: (27~~) P1’2 exp[z - (v’ - a)/221 (2 -+ a, Re(:) # 0) is used. This yields formally the Lagrangian in the path integral (12) mR’ L(a, q=y--h However, this expansion Re(z) #O and we explicitly problem as described by the deriving the corresponding
2(~+(D-3):2)2-1/4+gcothr, 2mR2 sinh’ c(
R
remains problematic, because it is only have Re(z) = 0. A justification that the Lagrangian (14) is the correct one can be time-dependent Schrodinger equation for
(14) valid for potential given by the path
216
CHRISTIAN
integral (12) with functional resulting in the Hamiltonian Hz--e..e
measure
GROSCHE
(13) (see [25]
h2 d= +h2(~+(D-3)/2)2-1/4 2mR2 da= 2mR2 sinh2 CI
for an explicit
-- ze2cothu. R
derivation),
(15)
An investigation of the behaviour a’, TV”-+ 0 in the short-time kernel of the path integral (12) shows that it is vanishing powerlike in sinh ~1,whereas the use of the asymptotic expansion produces exponential-like vanishing cc e ~l’slnha for CL+ 0 which is far too fast. This is very similar to the case of the radial path integral as discussed by Steiner [43]. The potential problem of Eqs. (14), (15) describes the Manning-Rosen potential [32] (A and B constants)
p+fR’(r) =
B
sinh’ (r/R)
- A coth f
(16)
which has been discussed by path integrals in Ref. [22]. It has been calculated with the help of the path integral for the modified Poschl-Teller potential which in turn is based on the SU(1, 1) path integral [S]. We have, in the functional measure interpretation, j Dr(t)
exp [f j!:’ (y i2 - sinh2:r,R)
+ A coth i)
dt]
(17)
Note that even the coth(r/R) term remains problematic (as the l/r potential in a radial path integral which has been pointed out by Kleinert [29]). It needs a regularization in a similar manner as the l/r2 centrifugal-term-see above. But this will not be discussed here. The discrete wave-functions and energy levels are given by
x 2F1
-n,2k,-n-l;s-tl;--
1 1-U >
(18)
KEPLER
PROBLEM
ON
4mAR P(n+2s+
N’MR’ --n
THE
217
PSEUDOSPHERE
1 112 1)2+x
(2K-l)~(k,+k,-ti)~(k,+k~+ti-l) T(k,-k,+ti)f(k,-k2-rc+l)
li2
1
1)’ 2mA2R2 E’MR’ = m/j? (s+2n+ n 8mR2 - h2(s + 2n + 1)”
(19)
where s+2n+l 2
+
2mAR’ ii’(s + 2n + 1 ) 1
k2=f(l+/~)+l+i)
(20)
k-=k,-k,--n, U= The continuous y(MR) P
1 - coth(r/R) 2
states are given by _ N(MR)~-~P’Z
(u-
l)r~12-(l+s)!?
P
l+s+i(fi-P)
l+s-i(d+P);s+l,
1 ‘1-U -)
(21)
T(k”,+k~-R)I--/?,+k2+I;.) x f(E, + k, + R - 1) r( -z,
+ k, - R + 1 )] ‘I’,
where C = i( 1 + ip), li”I = f( 1 + ip), d = 2mR’(E, - A)/h, and E, = ri2p2/2mR2 - A. Using the result of the ManninggRosen potential we identify f& which
,=(l+(D-3)i2)2-lj4 2mR2
R’
gives for the parameters k,=f(l+R+$),
’
(22)
k, , k,, K, and s, k,=I+y, (23)
s=21+D-3,
.q1+-$-w),
where fi= N + (D - 4)/2 with the principle quantum 1, .... N,W, the Bohr radius a=fz’/mZe’, and N, < &
number N = n + I+ 1 = - (D - 4)/2. Therefore I
218
CHRISTIAN
GROSCHE
finally obtain the path integral solution of the generalized Kepler problem pseudosphere zcKepler’( { ,I’}, { W}; T) = f f [ F e-‘~“=‘v$Ju’, I=0 p=t ?7=0
on the
Q’) Yn,,,,(u”, cl”)
+ jam dp e~iEPT’hY~,I.II(u’, Q’) !Yp,l.p(~“, Q”)
1 .
(24)
The bound state wave functions are given by 2’+(D-2)/2~(3-D))/2
(2j+D-3)!
y%LP(~~Q)=
fl x[(&-jp)
(N+1+D-4)!T(R/afi+(D-2)/2+f) (N-l-l)! I-(R/a&Z-(D-4)/2) R -z+l+D-3-N aN
x sinh’ a exp
2F,
1
>I
D-2
x
1’2
2
R
-N+I+1,1+2+--&21+D-2;
1 + coth c1
S?(Q)? (25)
with the energy-levels f12 - ((D - 2)/2)2 mZ2e4 -m. 2mR2
The continuous
(26)
states read 2(i/2)(P--P)+I+(D-22)/2
qd,,(4
Q) =
n(21+ D-
3)!
E-;(d+P)] >I 2
xsinh’aexp
[(
c1 k(fi+p)-Z-1
D-2
l+y+;(j-p),l+y-;(j+p); 21+D-2;
2
1 + coth u
V(Q).
(27)
with the energy spectrum (28)
KEPLER
PROBLEM
ON
THE
219
PSEUDOSPHERE
with the largest lower bound E, =h2(D - 2)2/8mR2. This feature which is independent of the specific potential here, appears also for the free motion on the pseudosphere and has been already discussed in Ref. [25]. For the discrete spectrum I obtain, in particular, for D = a’+ 1 = 4,
x sinh’a exp [_ x
--c1 -$+/+1-N (
>I 2 yw) 1 + coth u >
-Nfl+l,1+1+$;21+2;
2F,
(29)
with energy-levels
These are the results of Refs. [7, 281. For a reasonable radius of order 1 x 10”-light years we obtain Nz 1.5 x lOI which is an number. The highest level lies between -3h’/2mR2 and h2/2mR’ negative or positive [28]. Let us consider the limit R + ix), i.e., the flat space limit. First
the universe R of enormously large and thus may be of all we see that
mZ 2e4 EN = - 2h2(N+ (d- 3)/2)2 which are the well known Bohr levels. Together geometric function [ 3 1, p. 2621 lF,(a;c;z)=lim,,,
with the property
(31) of the hyper-
2F,(a,b;c;4b),
(32)
we get for the various terms
>I
-&+[+D-3-N 1 ’ (21+D-3)!
(N-l=(N+l+d-3)!
lim 2F, R-K
1Y L,2,+d-2j N-‘-’
=e-““’
D-2 [+I--N,l+~+;;~~;21+0-2;
R
2 1 + coth c(
2r ( 27 >
(33)
220
CHRISTIAN
GROSCHE
which give the bound state wave functions for the l/r potential
yn,l,p(r, Q) =z [ N2
(N-i-
l)!
]“‘(~)‘($)‘“-“‘*
(N+l+d-3)!
a(N+
in d dimensions,
a3
(;-3)/z))
Ljyz’l%2)
For the continuous states we have, similarly limit properly we must set p -pR),
= (-2ipr)~‘-d’J2~1e~iP’Mi,ap,,+~d-22),2(
(a(Nf
:d’- 3)/z)) sr(sz)’
(34)
(note that in order to perform the
-2ipr).
This gives, up to a phase factor, the continuous d-dimensional l/r potential,
state wave functions
of the
,.(I-d/2 yp’i’(r~R)=~(2*+d-2)!r ’
Mi,op,I+(d~2),2(-2ipr)
(36)
s?(Q),
and the energy-spectrum E, = h2p2/2m. These wave functions are, up to the constant factor en’2ap, correct. The feature of the missing factor in the limit of R + co was also observed in the case of the Kepler problem in the space of constant positive curvature [S]. The flat space limit can, of course, also be recovered by considering R + co directly in the path integral (12). For Z = 0 we easily find the continuous wave functions on the D-dimensional pseudosphere [4,25]: y$j(p,
52)= R(1-o)‘2
r(@+‘;(z-2)‘2)
(sinh u)(~-~)/~
pjir,9:/’
(co& ,r~)S;(Q). (37)
KEPLER
221
ON THE PSEUDOSPHERE
PROBLEM
Here Pr denote Legendre functions [31, p. 1583:
and use has been made of the representation
(38) The energy spectrum
is, of course, the same as in Eq. (28).
IV.
SUMMARY
In this paper I have discussed the path integral for the potential problem I’(a) = - (Ze”/R) x (coth a - 1) on the D-dimensional pseudosphere which I have also called the generalized Kepler problem on A”- ‘. The calculation was performed with the help of the path integral identity developed in an earlier work which made it possible to separate the angular variables from the hyperbolic one. The remaining path integral turned out to be equivalent with the path integral problem for the ManninggRosen potential. The known path integral solution of this potential could be easily applied, thus giving the complete path integral of the Kepler problem on the pseudosphere. Wave functions and energy spectrum have been explicitly stated and it was found that the corresponding limits for the flat space and for the free motion on the pseudosphere are in accordance with the known results. The bound-state energy levels for, e.g., D = 4 (d= 3, respectively) are given by N’1 mz’e” E,+-hpm-
2mR2
2h2N'
(N=O, 1, .... N,
Note added in proqf: and Junker [6], dealing
After with
acceptance of the manuscript, the same subject.
there was a publication
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155
by Barut,
Inomata,
222
CHRISTIAN
GROSCHE
4. M. BANDER AND C. ITZYKSON, Rev. Mod. Phys. 38 (1968), 330, 346. 5. A. 0. BARUT, A. INOMATA, AND G. JUNKER, J. Phys. A: Math. Gen. 20 (1987), 6271. 6. A. 0. BARUT, A. INOMATA, AND G. JUNKER, J. Phys. A: Math. Gen. 23 (1990), 1179. 7. A. 0. BARUT AND R. WILSON, Phys. L&r. A 110 (1985), 351. 8. M. B~HM AND G. JUNKER, Phys. Left. A 117 (1986), 375; J. Math. Phys. 28 (1987), 1978. 9. L. CHETOUANI AND T. F. HAMMANN, J. Math. Phys. 27 (1986), 2944. 10. E. D.‘HOKER AND D. H. PHONG, Nucl. Phys. E 269 (1986), 20; Comm. Math. Phys. 104 (1986). 537. 11. B. S. DE WITT, Rev. Mod. Phys. 29 (1957), 377. 12. J. C. D’OLIVIO AND M. TORRES, J. Phys. A: Murh. Gen. 21 (1988), 2351. 13. A. ERDELYI, W. MAGNUS, F. OBERHETTINGER, AND F. G. TRICOMI (Eds.), “Higher Transcendental Functions,” McGraw-Hill, New York, 1985. 14. R. P. FEYNMAN, Rev. Mod. Phys. 20 (1948), 367. 15. J. L. GERVAIS AND A. JEVICKI, Nucl. Phys. B 110 (1976), 33. 16. G. GILBERT, Nucl. Phys. B 277 (1986), 102. 17. I. S. GRADSH~YN AND I. M. RYZHIK, “Table of Integrals, Series, and Products,” Academic Press, New York, 1980. 18. M. B. GREEN, J. H. SCHWARZ, AND E. WITTEN, “Superstring Theory I, II,” Cambridge Univ. Press, Cambridge, 1987. 19. C. GROSCHE, Phys. Let/. A 128 (1988), 113. 20. C. GROSCHE, Fortschr. Phys 38 (1990), 531. 21. C. GROSCHE, Ann. Phys. (N.Y.) 187 (1988). 110; Ann. Phys. (N.Y.) 201 (1990), 258. 22. C. GROSCHE, J. Phys. A: Math. Gem 22 (1989), 5073. 23. C. GROSCHE AND F. STEINER, 2. Phys. C 36 (1987), 699. 24. C. GROSCHE AND F. STEINER, Phys. Left. A 123 (1987), 319. 25. C. GROSCHE AND F. STEINER, Ann. Phys. (N.Y.) 182 (1988), 120. 26. M. C. GUTZWILLER, Phys. Scripfa 9 (1985), 184. 27. L. INFELD, Phys. Reu. 59 (1941), 737. 28. L. INFELD AND A. SCHILD, Phys. Rev. 67 (1945), 121. 29. H. KLEINERT, Phys. Left B 224 (1989), 313. 30. T. D. LEE, “Particle Physics and Introduction to Field Theory,” Harwood Academic, New York, 1981. 31. W. MAGNUS, F. OBERHETTINGER, AND R. P. SONI, “Formulas and Theorems for the Special Functions of Theoretical Physics,” Springer-Verlag. Berlin, 1966. 32. M. F. MANNING AND N. ROSEN, Phys. Rev. 44 (1933). 953. 33. M. S. MARINOV, Phys. Rep. 60 (1980), 1. 34. I. M. MALES AND J. S. DOWKER, Proc. Roy. Sot. London Ser. A 327 (1972), 131. 35. D. C. MCLAUGHLIN AND L. S. SCHULMAN, J. Malh. Phys. 12 (1971), 2520. 36. M. M..MIZRAHI, J. Math. Phys. 16 (1975). 2201. 37. M. A. NAMAZIE AND S. RAJEEV, Nucl. Phys. B 277 (1986), 332. 38. M. OMOTE, Nucl. Phys. B 120 (1977), 325. 39. K. PAK AND I. SGKMEN, Phys. Left. A 103 (1984), 298. 40. A. M. POLYAKOV, Phys. Lefr. B 103 (1981), 207. 41. E. SCHR~DINGER, Proc. Roy. Irish Sot. 46 (1941), 9, 183; 47 (1941), 53. 42. M. SIEBER AND F. STEINER, DESY Preprint DESY 89-003. 43. F. STEINER, “Bielefeld Encounters in Physics and Mathematics VII; Path Integrals From meV to MeV, 1985,” p. 335, (M. C. Gutzwiller et a/. Eds.), World Scientific, Singapore, 1986. 44. A. F. STEVENSON, Phys. Rev. 59 (1941). 842. 45. H. A. VILENKIN, “Special Functions and the Theory of Group Representations,” Amer. Math. Sot., Providence, RI, 1968.
OF PHYSICS
204, 208-222
(1990)
The Path Integral for the Kepler Problem on the Pseudosphere CHRISTIAN
Technology
The Blackett and Medicine,
GROSCHE*
Laboratory, Imperial College of Science, Prince Consort Road, London SW7. United Received
February
Kingdom
20, 1990
The path integral for the generalized Kepler problem on the pseudosphere is calculated. As the generalized Kepler problem we describe the potential problem P’(a) = - (Ze*/R)( coth G(- 1) (a >O, R-curvature) on the D-dimensional pseudosphere, whereas for the genuine Kepler problem this feature of the potential is only true for the four-dimensional pseudosphere. Energy spectrum and wave-functions are explicitly calculated. The result is compared with the flat-space limit (i.e., R + “c) and with the free motion on the pseudosphere. The application of the result of the path integral solution of the Manning-Rosen potential plays a crucial role in the calculation. f’ 1990 Academic Press, Inc.
I. INTRODUCTION In this paper I study a specific potential problem on the D-dimensional pseudosphere, the “generalized Kepler problem.” The pseudosphere is a space of constant negative curvature. Spaceslike this have recently become important in the study of strings and quantum chaos. Whereas in the former it turns out that Laplacians on bounded domains in these spacesare relevant in the partition function 2 of the Polyakov [41] approach to string theory [ 10, 16, 18,371, in the latter one studies the dynamical properties of classical and quantum motion in these spaces [ l-3,26,42]. However, also quantum mechanical considerations on these spaces are important, especially calculations by path integrals, where the emphasis lies in the attempt to build up quantum mechanics on an equal footing, on the one hand, by operator calculus and, on the other, by Feynman’s path integration [14]. In an earlier paper we have already studied the free motion on the pseudosphere [25], including its connection in the three-dimensional case to the Poincare upper half-plane [24] and the Poincare disc [20]. Further contributions are due to Gutzwiller [26] and Bohm and Junker [S]. Also the effect of including a magnetic field was studied [21]. * Supported
by Deutsche
Forschungsgemeinschaft
under
208 0003-4916/90
$7.50
Copyright Ci 1990 by Academic Press, Inc. All rights of reproductmn in any form reserved.
Contract
DFG
Gr 1031
KEPLER PROBLEMON
THE PSEUDOSPHERE
209
The study of the Kepler problem is quite old and was motivated to compare the properties of the Coulomb potential in an “open hyperbolic” universe to that of an “open but flat” universe. As Infeld and Schild [28] found, the main difference is simply the fact that in an open hyperbolic universe there are only a finite (but very large) number of bound states. The same, of course, can be done for a closed universe, a problem first discussedby Schrijdinger [41]. Other attempts to the latter problem are due to Infeld [27], Stevenson [44], and Barut and Wilson 171, whereas the path integral treatment is due to Barut, Inomata, and Junker [S]. The further content of this paper is as follows: In the next section I construct the path integral on the D-dimensional pseudosphere /1” ‘. In Section III the generalized Kepler problem is formulated and its path integral is calculated. Whereas the notion “Kepler problem” is only true in the strict sensefor the D = 4 case, it can be easily generalized to the general D case, because I’ depends only on the variable LX.Due to this property the angular variables can be easily separated, leading eventually to the path integral problem for the Manning-Rosen potential which has already been solved in Ref. [22]. Using this solution, the path integral for the generalized Kepler problem on the pseudosphereis immediately obtained. In the flat space limit (i.e., R + t-x) we can recover the result of the I/r-potential problem in n= D - 1 dimensions for the discrete and continuous state wavefunctions and energy-spectra, respectively. For the case of vanishing potential strength the known results of the free motion on the pseudosphere are obtained. The last section contains some concluding remarks.
II. CONSTRUCTING
THE PATH INTEGRAL
Let us sketch the canonical way to construct a path integral on a curved manifold [ll, 12, 15, 23, 30, 33-36, 381. One considers the generic case, where the classical Lagrangian is given by
(1) with the metric tensor g,, and line elements d.?= guhdqudqh. The quantum Hamiltonian reads
where A,, is the LaplaceeBeltrami operator, g = det( gab) and g”’ = (go,,) ‘. One considers momentum operators
(3)
210
CHRISTIAN
which are hermitian
GROSCHE
with respect to the scalar product (fi,fi)
= j” Ji
dqf3q)fAq).
(4)
The crucial point in the construction of the path integral is the question of which ordering prescription one should use in the Hamiltonian. One well-known ordering rule is the Weyl-ordering which yields for the Hamiltonian
ff=&
(g”b~,pb+2p,g”bpb+~,pbg”b)+
with the well-defined
quantum A Vwey, = &
=r(iii
v(q
I+ A Vweydq
(5)
1
potential ( gabT;J;,-
[fbrorb
R’)
+
2k”brb),,
+
(6)
fb,ubl.
Here R’ denotes the scalar curvature and r;, are the Christoffel symbols. The resulting path integral constructed from the Hamiltonian (5) is then given in the mid-point lattice description:
K(q”, q’; T) = [g(q’)
g(q”)]
x exp
{I
-‘I4 N.-(~)ND’2(~~,‘Idy(j)),~~~ lim
5 2 g,,(q”‘)
Aq”‘“Aq(j’b
- cV(q(j’)
-I>
- &A V(q(j))
,
(7)
where q(j) = q(t’ +jE), g(l) = $($- 1) + qlj)), Aq(A = q(A _ q(j- I), E = T/N, T= f _ tt in the limit N + co. (I use the symbol A for different purposes; however, the relevant meaning should be clear by context.) An alternative approach is a product ordering [ 191 of the quantum Hamiltonian H. Here it is assumed that g,, can be written as g,, = h,,h,,; then ff=$f%,p,hcb+ with the well-defined AV,,
quantum = AVwey, +;
V(q)+AV,,(q)
(8)
potential (2hnchbr.ab - har,ohbr,b - hac,bhbc,a).
(9)
KEPLER
The resulting
PROBLEM
ON
THE
PSEUDOSPHERE
211
path integral reads
K(q”, q’; T) = N-r lim (&)ND~2‘~~,‘j&i??4~~’
-&V(q”‘)-edI/,,(q”‘) In order to specify our problem we must define g,, and V. Whereas P’ will be fixed in the next section, I show now the construction of g,, and the corresponding path integral on the pseudosphere. I introduce D-dimensional pseudospherical polarcoordinates [45] (see also [25] for some more details), x,=Rcoshcc x2 = R cash M cos 0,
2
sz=RcoshcrsinO,
,cosODP,
(11)
s D ,=RcoshccsinODP,...sinHzcos~
xD = R cash tl sin 0,
z . . sin I32sin 4,
where CI2 0, 0 < 8,, d 71(v = 2, .... D - 2) and 0 < 4 < 271(I also use for shorthand the notation 8,-, = a and 0, = 4). The metric tensor in x-space reads G,h= (- 1, 1, .... 1) (a, b = 1, .... D) such that .x?= -R’= -.~f + Cf=‘=, .x: with R fixed defines the D-dimensional pseudosphereAD ‘. The Gaussian curvature reads K= -(Dl)(D-2)/R’. The metric tensor on the pseudosphere is given by g,,= R’(1, sinh’ ~1, sinh’ c1sin’ 8,-,, .... sinh’ cx.. .sin’ 02) (a, b = 1, .... D- 1). Therefore, for the Hamiltonian
+...+
smh’ CX.. sin’ 8? &j
212
CHRISTIAN GROSCHE
with the quantum potential AY(O)=&[(D-2&y--
... -
1 sinh’ CI.. . sin2 e2I
(13)
({ 0) denotes a collection of variables). Due to the specific nature of g,, we have AV Wey,= AV,, E AV. Thus we obtain the (Lagrangian) path integral on the D-dimensional pseudosphere AD ~ I:
SinhD-2
cl(j)
&(j)
&J(J)
s x exp i ,g 1
[Lg,({e(j-
I)}, { e(jJ})-Ev({e(j)})-Edv({e(j)})]
J=I
.
(14)
I
L,, denotes the classical Lagrangian, [oi2+sinh2&iP2+
... +(sinh2cr...sin2e2)$2],
(15)
and its counterpart on the lattice reads
Lg,({e(j-l)), {e(q) n =$[A Here f2(q”‘)
2x(J)
=f(q”-
+
sinh2
E(i)A20(/)
’ ‘f(q(j’)
D-2
+
. .
n A24(j)] . + (sinh2 ~1~‘). . . sin2e(jl) 2
.
(16)
and D-2
dQ = n
(sin ek)k-l de,
(17)
k=l
denotes the (D - 2)-dimensional surface element on the unit-sphere SD-’ [ 13, Chap. XI]. This completes the construction of the path integral on the D-dimensional pseudosphere AD- ‘.
III. THE KEPLER PROBLEM The quantum mechanical Kepler problem in a space of constant negative curvature was first discussedby Infeld and Schild [28] by the factorization method. They introduced it to compare its energy spectrum of a hydrogen atom in an open
KEPLER
hyperbolic universe to that difference lies in the fact that levels is finite, however very sional hyperbolic space with
ds’ =
dr’
1 - r’/R’
+ r’(d0’
PROBLEM
213
ON THE PSEUDOSPHERE
in an open flat universe. We shall find that the main in an open hyperbolic universe the number of discrete large indeed. The line element of the (three-) dimenthe coordinates (r, 8,$) can be written in the form
+ sin’ 8 dq5’)
(~>0,HECO,7Cl,~ECO,27Cl), (1)
= R’[dc?+ sinh’ @do2 + sin’ 8 d@)]
where we have set sinh r = r/R. Note that the dimension of this d= 3 dimensional hyperbolic space corresponds to the D = 4 dimension of the pseudosphere n3. The potential energy V(x) of an electron in a central ‘Coulomb-like” field must satisfy Poisson’s equation A,,
V(m) = -47TZe%(a),
(2)
where d,, is the Laplace-Beltrami operator in the three-dimensional hyperbolic space. Thus we find for V, V(a) = - g if we demand that V+O
(coth x - 1),
as C(+(XCIand V-+
-Ze’/r
as R-+m
(R-tee,
aR=r
fixed).
Although this potential is in the strict senseonly Coulombian for D = d + 1 = 4 we can easily generalize it to arbitrary D or d, respectively. This is, of course, similar to the consideration of the l/r-potential problem in flat d-dimensional space [9], where only for d= 3 do we have the genuine Coloumb/hydrogen problem. In the following I will omit the d and D superscripts, respectively, and denote by the “generalized Kepler problem on the pseudosphere” the potential problem (3) on AD ~ ‘. Thus I obtain the path integral
.
(4)
d V( 10) ), of course, as in Eq. (II.1 3). Due to the very singular nature of d V( (0)) this path integral is at it stands is not tractable. However, we can use a path integral identity (based on a method developed in [ 15, 391) already derived in
214
CHRISTIAN
GROSCHE
Ref. [23] to simplify the path integration significantly and separate the angular variables BoPz, .... 4 from the hyperbolic coordinate CY.I introduce the quantity $(‘.” defined by
D 3 c cosfycosH),~’ rn= I
cos I) “~2’=cosOj:‘2COS8~‘r+
which is actually the addition theorem on the S”- ‘-sphere and cos $“.” = Q(“..Q’“, where Q’i.” are unit vectors on the S” ‘-sphere. Using the result of Ref. [23] the following path integral identity can be achieved (replace R = r”) = sinh cx”’ in Eq. (2.27)): exp
imR’ ci, 2Ehsmh-cr
[d’6)g)z+
1 :+ ... +sin2(j~~2...sin’(j~~
h*(D-2)’ 8mR’
Now expanding
e:cos$w 7 =
according
7 v : z-(v) f. (Ifv) z 0 /=O
[ 11 ]-to denote The highly singular
sinh”P’rDr(f)i’
[ti’ + 2 sinh” a( 1 - cos $)] + g
the exponential
(6)
>I
A Pfollowing Dewitt Here I have used the symbol “equivalence as far as use in the path integral is concerned.” terms are cancelling and I obtain
-
A” #(I)]
id 8mR’ sinh’ %“I
(1 -co~Ic/“~‘.“)-
sln2(jgJ
? . sin&$))
... +(sin/;;;i;:’
DO(r) coth c(
(7)
to [ 17, p. 9801, C;‘(cos
l+v’y
Z,+.(z),
(8)
together with [ 13, Chap. XI]
f Sf’(D”‘)S~(R”~)=2*(D1~l~~2r(~)2f~”33c~D~~~~~(cosiil-‘)), (9) p = 1
KEPLER
PROBLEM
ON
THE
215
PSEUDOSPHERE
where the S?(0) denote the real hyper-spherical harmonics of degree 1 with vector Q on the S” -- ’ -sphere, I~N~and~=l,...,M,M=(21+D-2)(1+D-3)!/ f!(D-2)!. Thus for v=(D-3)/2: e:cos*“~” = 2X 271 ( z> Therefore
(D-3)/2
% A4 c c s~(~“‘)s~(~“‘)z,+,,~~,,,2(=). /=o I’=1
I obtain for the path integral ~(Kep’er)( (o”},
{y};
unit
(10)
(7)
T) h’(D-2)’ 8mR2
!I
The kernel K, is given by
x
p,[sinh’
z] Dcc( t)
>1
a At , with the functional p,[sinh’
(12)
measure ~~[sinh’ E], I\ CX]= lim p,[sinh’ ~“‘1 N-r N
= lim n WV-.%,=I
znrnR’ /‘--A ~ smh’ tl(” i&h
x exp ~sinhC%)) (
“2
I,+,.~-,,,,
(gsinG/)).
(13)
TO get into contact with a one-dimensional potential problem the asymptotic expansion I,,(z) 2: (27~~) P1’2 exp[z - (v’ - a)/221 (2 -+ a, Re(:) # 0) is used. This yields formally the Lagrangian in the path integral (12) mR’ L(a, q=y--h However, this expansion Re(z) #O and we explicitly problem as described by the deriving the corresponding
2(~+(D-3):2)2-1/4+gcothr, 2mR2 sinh’ c(
R
remains problematic, because it is only have Re(z) = 0. A justification that the Lagrangian (14) is the correct one can be time-dependent Schrodinger equation for
(14) valid for potential given by the path
216
CHRISTIAN
integral (12) with functional resulting in the Hamiltonian Hz--e..e
measure
GROSCHE
(13) (see [25]
h2 d= +h2(~+(D-3)/2)2-1/4 2mR2 da= 2mR2 sinh2 CI
for an explicit
-- ze2cothu. R
derivation),
(15)
An investigation of the behaviour a’, TV”-+ 0 in the short-time kernel of the path integral (12) shows that it is vanishing powerlike in sinh ~1,whereas the use of the asymptotic expansion produces exponential-like vanishing cc e ~l’slnha for CL+ 0 which is far too fast. This is very similar to the case of the radial path integral as discussed by Steiner [43]. The potential problem of Eqs. (14), (15) describes the Manning-Rosen potential [32] (A and B constants)
p+fR’(r) =
B
sinh’ (r/R)
- A coth f
(16)
which has been discussed by path integrals in Ref. [22]. It has been calculated with the help of the path integral for the modified Poschl-Teller potential which in turn is based on the SU(1, 1) path integral [S]. We have, in the functional measure interpretation, j Dr(t)
exp [f j!:’ (y i2 - sinh2:r,R)
+ A coth i)
dt]
(17)
Note that even the coth(r/R) term remains problematic (as the l/r potential in a radial path integral which has been pointed out by Kleinert [29]). It needs a regularization in a similar manner as the l/r2 centrifugal-term-see above. But this will not be discussed here. The discrete wave-functions and energy levels are given by
x 2F1
-n,2k,-n-l;s-tl;--
1 1-U >
(18)
KEPLER
PROBLEM
ON
4mAR P(n+2s+
N’MR’ --n
THE
217
PSEUDOSPHERE
1 112 1)2+x
(2K-l)~(k,+k,-ti)~(k,+k~+ti-l) T(k,-k,+ti)f(k,-k2-rc+l)
li2
1
1)’ 2mA2R2 E’MR’ = m/j? (s+2n+ n 8mR2 - h2(s + 2n + 1)”
(19)
where s+2n+l 2
+
2mAR’ ii’(s + 2n + 1 ) 1
k2=f(l+/~)+l+i)
(20)
k-=k,-k,--n, U= The continuous y(MR) P
1 - coth(r/R) 2
states are given by _ N(MR)~-~P’Z
(u-
l)r~12-(l+s)!?
P
l+s+i(fi-P)
l+s-i(d+P);s+l,
1 ‘1-U -)
(21)
T(k”,+k~-R)I--/?,+k2+I;.) x f(E, + k, + R - 1) r( -z,
+ k, - R + 1 )] ‘I’,
where C = i( 1 + ip), li”I = f( 1 + ip), d = 2mR’(E, - A)/h, and E, = ri2p2/2mR2 - A. Using the result of the ManninggRosen potential we identify f& which
,=(l+(D-3)i2)2-lj4 2mR2
R’
gives for the parameters k,=f(l+R+$),
’
(22)
k, , k,, K, and s, k,=I+y, (23)
s=21+D-3,
.q1+-$-w),
where fi= N + (D - 4)/2 with the principle quantum 1, .... N,W, the Bohr radius a=fz’/mZe’, and N, < &
number N = n + I+ 1 = - (D - 4)/2. Therefore I
218
CHRISTIAN
GROSCHE
finally obtain the path integral solution of the generalized Kepler problem pseudosphere zcKepler’( { ,I’}, { W}; T) = f f [ F e-‘~“=‘v$Ju’, I=0 p=t ?7=0
on the
Q’) Yn,,,,(u”, cl”)
+ jam dp e~iEPT’hY~,I.II(u’, Q’) !Yp,l.p(~“, Q”)
1 .
(24)
The bound state wave functions are given by 2’+(D-2)/2~(3-D))/2
(2j+D-3)!
y%LP(~~Q)=
fl x[(&-jp)
(N+1+D-4)!T(R/afi+(D-2)/2+f) (N-l-l)! I-(R/a&Z-(D-4)/2) R -z+l+D-3-N aN
x sinh’ a exp
2F,
1
>I
D-2
x
1’2
2
R
-N+I+1,1+2+--&21+D-2;
1 + coth c1
S?(Q)? (25)
with the energy-levels f12 - ((D - 2)/2)2 mZ2e4 -m. 2mR2
The continuous
(26)
states read 2(i/2)(P--P)+I+(D-22)/2
qd,,(4
Q) =
n(21+ D-
3)!
E-;(d+P)] >I 2
xsinh’aexp
[(
c1 k(fi+p)-Z-1
D-2
l+y+;(j-p),l+y-;(j+p); 21+D-2;
2
1 + coth u
V(Q).
(27)
with the energy spectrum (28)
KEPLER
PROBLEM
ON
THE
219
PSEUDOSPHERE
with the largest lower bound E, =h2(D - 2)2/8mR2. This feature which is independent of the specific potential here, appears also for the free motion on the pseudosphere and has been already discussed in Ref. [25]. For the discrete spectrum I obtain, in particular, for D = a’+ 1 = 4,
x sinh’a exp [_ x
--c1 -$+/+1-N (
>I 2 yw) 1 + coth u >
-Nfl+l,1+1+$;21+2;
2F,
(29)
with energy-levels
These are the results of Refs. [7, 281. For a reasonable radius of order 1 x 10”-light years we obtain Nz 1.5 x lOI which is an number. The highest level lies between -3h’/2mR2 and h2/2mR’ negative or positive [28]. Let us consider the limit R + ix), i.e., the flat space limit. First
the universe R of enormously large and thus may be of all we see that
mZ 2e4 EN = - 2h2(N+ (d- 3)/2)2 which are the well known Bohr levels. Together geometric function [ 3 1, p. 2621 lF,(a;c;z)=lim,,,
with the property
(31) of the hyper-
2F,(a,b;c;4b),
(32)
we get for the various terms
>I
-&+[+D-3-N 1 ’ (21+D-3)!
(N-l=(N+l+d-3)!
lim 2F, R-K
1Y L,2,+d-2j N-‘-’
=e-““’
D-2 [+I--N,l+~+;;~~;21+0-2;
R
2 1 + coth c(
2r ( 27 >
(33)
220
CHRISTIAN
GROSCHE
which give the bound state wave functions for the l/r potential
yn,l,p(r, Q) =z [ N2
(N-i-
l)!
]“‘(~)‘($)‘“-“‘*
(N+l+d-3)!
a(N+
in d dimensions,
a3
(;-3)/z))
Ljyz’l%2)
For the continuous states we have, similarly limit properly we must set p -pR),
= (-2ipr)~‘-d’J2~1e~iP’Mi,ap,,+~d-22),2(
(a(Nf
:d’- 3)/z)) sr(sz)’
(34)
(note that in order to perform the
-2ipr).
This gives, up to a phase factor, the continuous d-dimensional l/r potential,
state wave functions
of the
,.(I-d/2 yp’i’(r~R)=~(2*+d-2)!r ’
Mi,op,I+(d~2),2(-2ipr)
(36)
s?(Q),
and the energy-spectrum E, = h2p2/2m. These wave functions are, up to the constant factor en’2ap, correct. The feature of the missing factor in the limit of R + co was also observed in the case of the Kepler problem in the space of constant positive curvature [S]. The flat space limit can, of course, also be recovered by considering R + co directly in the path integral (12). For Z = 0 we easily find the continuous wave functions on the D-dimensional pseudosphere [4,25]: y$j(p,
52)= R(1-o)‘2
r(@+‘;(z-2)‘2)
(sinh u)(~-~)/~
pjir,9:/’
(co& ,r~)S;(Q). (37)
KEPLER
221
ON THE PSEUDOSPHERE
PROBLEM
Here Pr denote Legendre functions [31, p. 1583:
and use has been made of the representation
(38) The energy spectrum
is, of course, the same as in Eq. (28).
IV.
SUMMARY
In this paper I have discussed the path integral for the potential problem I’(a) = - (Ze”/R) x (coth a - 1) on the D-dimensional pseudosphere which I have also called the generalized Kepler problem on A”- ‘. The calculation was performed with the help of the path integral identity developed in an earlier work which made it possible to separate the angular variables from the hyperbolic one. The remaining path integral turned out to be equivalent with the path integral problem for the ManninggRosen potential. The known path integral solution of this potential could be easily applied, thus giving the complete path integral of the Kepler problem on the pseudosphere. Wave functions and energy spectrum have been explicitly stated and it was found that the corresponding limits for the flat space and for the free motion on the pseudosphere are in accordance with the known results. The bound-state energy levels for, e.g., D = 4 (d= 3, respectively) are given by N’1 mz’e” E,+-hpm-
2mR2
2h2N'
(N=O, 1, .... N,
Note added in proqf: and Junker [6], dealing
After with
acceptance of the manuscript, the same subject.
there was a publication
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