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. 122
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?
(4.2)
Ga = t?a + µxa u?u ?u = , u?u , D = 2t?t + xa ?a + ?u?u ,
?u
µx2 n
A = tD ? t2 ?t + ?=?
u?u .
2 2
The Schr?dinger equation
o
(4.3)
2im?t + ?aa = 0,
Second-order differential invariants of the rotation group O(n) 517

? = ?(t, x) being a complex-valued function, is also invariant [16] under the genera-
lized Galilei algebra with the basis operators
? ?
, Jab = xa pb ? xb pa , J = i(??? ? ? ? ??? ),
, pa = ?i
p0 = i
?t ?xa
Ga = tpa ? mxa J, D = 2tp0 ? xa pa + ?I (I = ??? + ? ? ??? ), (4.4)
mx2 n
A = t p0 ? txa pa + ?tI + ?=?
2
J .
2 2
The asterisk means the complex conjugation.
We shall designate the algebra (4.4) with the symbol AGII (1, n). Besides,
2

AGI (1, n) = ?t , ?a , u?u , Ga , Jab ,
the operators being of the form (4.2). A basis of the algebra AGI (1, n) consists of
1
the basis operators or AGI (1, n) and of the operator D. Furthermore AGII (1, n) =
p0 , pa , J, Jab , Ga (4.4). A basis of the algebra AGII (1, n) consists of the previous
1
operators and also D (4.4).
To simplify the form of invariants, we introduce the following change of dependent
variables:
Im ?
(4.5)
u = exp ?, ? = exp ? Im ? = arctg .
Re ?
All the indices k in the expressions of the type (0.3) here will take on values from
1 to n, the indices j will take on values from 0 to k.
We seek invariants of the algebra AGI (1, n) in the form
2

(4.6)
F = F (?t , ?a , ?tt , ?at , ?ab ).
Obviously, they do not include ?, xa , and t because the basis (4.2) contains operators
?? , ? a , ? t .
Using the definition of an absolute differential invariant (0.2) we get the following
conditions on the function F (4.6):
2
J ab F = ?a F?b ? ?b F?a + F?bt ?at ? ?bt F?at + 2?ac F?bc ? 2?bc F?ac = 0, (4.7)
2
Ga F = ??a F?t + µF?a ? 2?at F?tt ? ?ab F?bt = 0, (4.8)
2
D F = ?2?t F?t ? ?a F?a ? 4?tt F?tt ? 3?at F?at ? 2?ab F?ab = 0, (4.9)
2 2 2
A F = t D F + xa Ga F ? ?F?t ? 2?t F?tt ? ?a F?at + µ?ab F?ab = 0. (4.10)
From equations (4.8), we can see that the tensors
(4.11)
?a = µ?at + ?b ?ab , ?ab
are covariant with respect to the algebra AGI (1, n) (µ = 0).
Theorem 9. There is a functional basis of absolute differential invariants for the
algebra AGI (1, n), when µ = 0, consisting of these 2n + 2 invariants:
M1 = 2µ?t + ?a ?a , M2 = µ2 ?tt + 2µ?a ?at + ?a ?b ?ab ,
(4.12)
Rk = Rk (?a , ?ab ), Sk = Sk (?ab ).
518 W.I. Fushchych, I.A. Yegorchenko

For the algebra AGI (1, n) (µ = 0) such a basis has the form
1

M2 Rk Sk
(4.13)
2, , .
2+k k
M1 M1
M1

For the algebra AGI (1, n) (µ = 0), there is a basis of the form
2

? ?
N2 Rk Sk
(4.14)
2, , (k = 2, . . . , n),
2+k k
N1 N1
N1
where
N1 = 2µ?t + ?a ?a + ?aa ,
1 1 1
N2 = µ2 ?tt + 2µ ?a ?a ?bb + ?2 ,
?t ?aa + ?a ?at + ?a ?b ?ab +
n bb
n n
k
(?n)l k! (4.15)
? k?1
Rk = Rl (?aa ) ,
l!(k ? l)!
l=0
k
(?n)l (k ? 1)!(k + 1)
? Sl (?aa )k?l ,
Sk =
(l + 1)!(k ? l)!
l=0

Sk , Rk are defined by (4.12) and ?a has the form (4.11).
The proof of this theorem is similar to the proof of Theorems 2 and 3. We shall
present here only some hints to the proof.
It is evident that the function F must depend on the invariants of the Euclid
algebra

F = F (?t , ?tt , Rk (?a , ?ab ), Rk (?at , ?ab ), S?ab )).

First we construct two invariants of AGI (1, n) M1 and M2 (4.12) which depend on ?t
and ?tt respectively. The other invariants of the adduced basis (4.12) do not depend on
?t or ?tt and the sets {M1 , M2 } and {Rk , Sk } are independent. The invariants Rk , Sk
are constructed with the covariant tensors ?a , ?ab (4.11) similarly to invariants of the
conformal algebra investigated above, and it is easy to see that they are independent.
The generic ranks of the prolonged algebras AGI (1, n), AGI (1, n), AGI (1, n) are
1 2
equal to the numbers of their operators and from this fact we can compute the number
of elements in the bases for these algebras.
Adding to (4.7) and (4.8) the condition (4.9), we obtain from the invariants (4.12)
the basis (4.13) for the algebra AGI (1, n).
1
? k , Sk (4.15) of the algebra AGI (1, n) were found from the
?
Relative invariants R 2
equation

?F?t ? 2?t F?tt ? ?a F?at + µ?ab F?ab = 0,

F = F (Rk , Sk ), and then we constructed absolute invariants using (4.9). Besides, it is
??
possible to construct analogues to Rk , Sk with AGI (1, n)-covariant tensors ?a (4.11)
2
and
2?ab
?ab = ?ab ? (?c ?c + µ?t ).
n
Second-order differential invariants of the rotation group O(n) 519

Considering (?at ), (?a ), (?ab ) as independent vectors and tensors and putting
?ab = 0 whenever a = b, ?a = 0, we see from Lemma 2 that the adduced sets of
invariants are independent.
Note 2. A basis of invariants for the Galilei algebra without translations contains
expressions (4.12) and
12
µx ? ?t,
Rk (ha , ?ab ),
2
the Galilei-covariant vector ha having the form
ha = µxa ? t?a .
Let us also adduce an A-covariant tensor
µxa
? ? ?a
ha =
t
depending on xa , and a relative invariant of the operators A and D (4.2)
µx2
exp ? ?
2t
with which it is possible to construct a basis of invariants for the algebra Ga , Jab , D,
A.
We have presented a method to find the bases of invariants for Lie algebras for
which Jab (1.1) are basis operators. Further, we shall adduce functional bases for the
algebras AGI (1, n) where µ = 0 and AGII (1, n) where µ = 0 or µ = 0. We omit
2 2
proofs because they are similar to proofs of the previous theorems.
It is evident from the conditions (4.7)–(4.10) that the case µ = 0 for the algebra
AGI (1, n) has to be specially considered. The tensors (?a ) and (?ab ) are covariant
2
with respect to this algebra; the tensor (?a ) involved in invariants is defined by an
implicit correlation
(4.16)
?bt = ?a ?ab .
Theorem 10. There is a functional basis of the second-order differential invariants
for the algebra AGI (1, n), where µ = 0, that has the form
M1 = ?t ? ?a ?a , M2 = ?tt ? ?at ?a ,
(4.17)
Rk = Rk (?a , ?ab ), Sk = Sk (?ab ).
The corresponding basis for the algebra AGI (1, n), where µ = 0 has the form
1
2
M1 Rk Sk
, , ;
k k
M2 M1 M1
for the algebra AGI (1, n), when µ = 0, it has the form
2
Rk Sk
, ,
M 1/2k M 1/2k
where Rk , Sk are defined by (4.17) and
M = (?t ? ?a ?a )2 + (?tt ? ?at ?a )(? + ?a ?b rab ).
Here, the matrix {rab } = {?ab }?1 ; ?a = rab ?bt are the same as in (4.16).
520 W.I. Fushchych, I.A. Yegorchenko

Note 3. It is possible to use, instead of M1 , M2 , the invariants

··· ···
?t ?1 ?n ?tt ?1t ?nt
··· ···
?1t ?11 ?1n ?1t ?11 ?1n
? ?
M1 = , M2 = ,
··· ··· ··· ··· ··· ··· ··· ···
··· ···
?nt ?n1 ?nn ?nt ?n1 ?nn

which have been found in [17] as the solution of the problem of finding the equations
invariant under the Galilei algebra when µ = 0.
Note 4. The invariants for the algebra Jab , Ga , J, D, A (4.2), where µ = 0, which
depend on xa , t, can be constructed with ?a , ?ab and the following covariant vector
ha 2 4 xb ?b ?a
?
ha = + t?a ?t + ,
t n n t
where ha = xb ?ab + t?at is covariant with respect to the operators Ga when µ = 0.
4.2. Let us proceed to describe the basis of the invariants for the algebra AGII(1, n).
2
Theorem 11. Any absolute differential invariant of order ? 2 for the algebras listed
below is a function of the following expressions:
(1) AGII (1, n), m = 0:

? + ?? , M1 = 2im?t + ?a ?a , M1 , ?

?
M2 = ?m2 ?tt + 2im?a ?at + ?a ?b ?ab , M2 ,
Sjk = Sjk (?ab , ?? ), Rk = Rk (?a , ?ab ),
1
ab
Rk = Rk (?a , ?ab ), Rk = Rk (?a + ?? , ?ab ),
?
2 3
a

the covariant tensors being ?a = ?im?at + ?b ?ab ;
(2) AGII (1, n), m = 0:
1

? ? l 3
M1 M2 M2Rk Rk Sjk
, 2, 2, (l = 1, 2), , ,
2+k k k
M1 M1 M1 M1 M1
M1
?
? + ?? when ? = 0, M1 e(2/?)(?+? ) when ? = 0;

(3) AGII (1, n), m = 0, ? = ? n :
2 2

? ?l ?3 ?
N1 N2 N2 Rk Rk Sjk
?
N1 e(?4/n)(?+? ) , ?, 2, 2, (l = 1, 2), , ,
2+k k k
N1 N1 N1 N1 N1
N1
where

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( 135 .)



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