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r
Here
n2 d
? ? = K 2 ? S 2 ? 2S3 + nS2 + + 2iS2 (1 ? w2 )1/2 ?
2
4 dw
(3.5)
1 n n
?2S1 + S3 ?
L3 + + S3 w ,
(1 ? w2 )1/2 2 2

where
d2 d
K 2 = (1 ? w2 ) ? 2w ?
dw2 dw
[L3 + (n/2 + S3 ) ? (n/2 + S3 )w]2 2
n (3.6)
? ? + S3 ,
1?w 2 2
w = cos ?.

The solutions of eq.(3.4) can be represented as an expansion in Jacobi polino-
mials [14] and eigenfunctions of the operator L3

F? (r)Pn/2+j,n/2+? (w) exp[?i(j ? ?)?],
k
(3.7)
?=
?

k
where Pn/2+j,n/2+? is the complete set of the normalized eigenfunctions of the
commuting operators J3 = L3 + S3 , S3 and K 2 (3.6) which correspond to the ei-
genvalues j, ? and ?k(k + 1), respectively, moreover

k? and k ?
n n n
are integers,
+? , +j +?
2 2 2
(3.8)
? = ?nsk , ?nsk + 1, . . . , nsk , nsk = min(s, k).

Using recurrent relations [14]

? ?µw
d
(1 ? w2 )1/2 k
+ P? µ (w) =
dw (1 ? w2 )1/2 (3.9a)
= [(k + µ )(k ? µ + 1)]1/2 P? µ ?1 (w),
k
512 W.I. Fushchych, A.G. Nikitin, W.M. Susloparow

? ?µw
d
(1 ? w2 )1/2 ? k
P? µ (w) =
dw (1 ? w2 )1/2 (3.9b)
= ?[(k ? µ )(k + µ + 1)]1/2 P? µ +1 (w),
k



and formulae for the matrices S1 and S2 in Gel’fand–Zeytlin basis [21], we come to
the following equations for radial function F? (w):

DF? (r) = r?2 A?? F? (r), (3.10)

where
k(k + 1) ? n2 /4
d2 2d
D = (? ? m ) + 2 + ?
2 2
(3.11)
,
r2
dr r dr

1 ? 2s
A?? = s(s + 1) ? 2? 2 + n? ??? ? ??? ,
2s
? = ? ± 1, (3.12)
??? = 0,
1/2
n n
= ? (s + ?)(s ? ? + 1) k + + ? k? ??+1
????1 = ???1? .
2 2

Since the matrix A?? is diagonalizable, system (3.10) can be reduced to the
system of noncoupled equations

DF? (r) = r?2 B? F? (r),
? sk ? ? (3.13)
F? = u?? F? ,
sk
where B? are the matrix A?? eigenvalues, which coincide with the roots of the
characteristic equation

det A?? ? B? ???
sk
(3.14)
= 0,

u?? is the operator diagonalizing the matrix A?? .
Each of eqs.(3.13) by the replacement of the variable ? = (?2 ? m2 )1/2 r reduces to
the wen-known one [14]

? ? k(k + 1) ? n2 /4 + B? ?
d2 F sk
2 dF
+ 1? (3.15)
+ F = 0,
d?2 ?2
? d?

the solution of which (limited at the point ? = 0) is expressed via ssl’s function

1
F = v Jv(k+n/2+1/2)(k?n/2+1/2)+B sk (?),
? (3.16)
? ?



where k satisfies of the conditions (3.8).
One can make sure by the direct verification that at least for s < 3 , (k + n/2 +
2
1/2)(k ? n/2 + 1/2) + B? > 0. This means that ? > m, and so particle with spin
sk

s < 3 in magnetic-pole field has continuous energy spectrum and has not coupled
2
states. In the ses s = 0 and s = 1 the absence of coupled states was demonstrated
2
by Dirac [12] and Harish-Chandra [14].
Relativistic particle of arbitrary spin 513

According to the above the explicit solution of the wave eq.(2.4) for the case in
which the external field source is a magnetic monopole has the form
N
?s (t, r) = exp[?i?t] exp[i(S2 cos ? ? S1 sin ?)?]?
(?2 ? m2 )1/2 r (3.17)
(?)Jv(k+n/2+1/2)(k?n/2+1/2)+B sk (
?)?]u?1 Pn/2+j,n/2+?
? exp[?i(j ? ?2 ? m2 · r),
k
??
?


where N is the normalization constant. Solutions of the starting system (2.1) may be
expressed through the function (3.17) with the help of the relations (2.3c).
Let us give the explicit expressions for B? and usk , if s ? 1:
sk
??
1/2
1 n1 n1
1
2k
=± k? +
B k+ + ,
±1 4 2 2 2 2
2



?p/3 cos[(? + ??)/3],
1k
B? = 2
q 3 1
p = ?(2k + 1)2 + n2 ? ,
cos ? = ,
?(p/3)2 4 3
2
8 16
q = ? k(k + 1) ? , ? = 0, ±1;
3 27
?c1 1
1 c1
k
?, ? = ± ;
2
u?? = ,
c2 c2 2
? ?
p1 p2
? ?1 ?
1k 1 1k ? n/2 1
? ?
n/2 + B1 B1
? ?
? ?
p1 p2
=? ?, ?, ? = 0, ±1;
? ?2 ?2
u1k ? ?
1k 2 B0 ? n/2
1k
?? n/2 + B0
? ?
? ?
p1 p2
? ?3 ?3
1k 3 B?1 ? n/2
1k
n/2 + B?1
1/2 1/2
n n1 n n1
p1 = ? 2 k ? p2 = ? 2 k + k? +
k+ + , .
2 2 2 2 2 2

ere c1 , c2 , ?1 , ?2 , ?3 are arbitrary nonzero constants.

4. Arbitrary spin particle in the Coulomb field
In the case of Coulomb potential A0 = Ze/r, A = 0, eq.(2.4) in spherical co-
ordinates takes the following form:
2
1 ? 2? 1 ? i?
? m2 ? = ? (S · r)?, (4.1)
r ? + 2 ?? + ?+
r2 ?r ?r sr3
r r
where ? = Ze2 , ? is an angular part of Laplace operator.
Equation (4.l), as eq.(3.2), has exact solutions in separated variables. In [9] eq.(4.1)
is solved by using of the spherical spinor basis. Here we shall obtain the expressions
of eq.(4.1) solutions through Jacobi polinomials which are more convenient basis in
more general case of combination of the Coulomb and magnetic-pole potentials.
514 W.I. Fushchych, A.G. Nikitin, W.M. Susloparow

In such a way, as was done in previous section, we shall pass to the representation,
in which the matrix S · r/r3 is diagonal. Using for this purpose the transformation
operator (3.3), we obtain
2
1 ? 2? 1 ? i?
? + 2? ? ? + ? m2 ? = ? (4.2)
r ?+ S3 ?,
n=0
2 ?r sr2
r ?r r r
where ? ? is the operator (3.5) with n = 0.
n=0
Representing the solutions of eq.(4.2) in the form (3.7) one comes to eq.(3.10) for
the radial wave function, where
d2
2
? 2d k(k + 1)
? m2 + ? (4.3)
D = ?+ + ,
dr2 r2
r r dr
i?
A? = s(s + 1) ? 2? 2 ? ? ??? ? ?? , (4.4)
?? ??
s
?? is the matrix (3.12) corresponding to n = 0.
??
The matrix A? is diagonalizable, so the system of equations (3.10), (4.3), (4.4) is
??
sk
equivalent to noncoupled eqs.(3.13), (4.3), where B? are the roots of the matrix (4.4)
sk
characteristic equation (for explicit expressions for the coefficients B? see [9]). Each
of these equations in its turn is reduced to the well-known equation of the for [22]
d2 y dy l2
z
+ ?? ? (4.5)

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