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1 1
>W = >
W=
(xa ? t?a )2 + xb xb
xc xc
1
>W = > ···
(xa ? t(?a + ?a ))2 + xb xb

(iii) QA for A(t) = t or A(t) = t2 /2 do not change the potential, i.e.,

1 1 1
>W = >W = > ···
W=
xc xc xc xc xc xc

3 The Schr?dinger equation and conditions
o
for the potential
Consider several examples of the systems in which one of the equations is equation (2)
with potential W = W (t, x ), and the second equations is a certain condition for the
potential W . We ?nd the invariance algebras of these systems in the class of operators

X = ? µ (t, x, ?, ? ? , W )?xµ + ?(t, x, ?, ? ? , W )?? +
+ ? ? (t, x, ?, ? ? , W )??? + ?(t, x, ?, ? ? , W )?W .
280 W.I. Fushchych, Z.I. Symenoh, I.M. Tsyfra

(i) Consider equation (2) with the additional condition for the potential, namely
the Laplace equation.
??
i + ?? + W (t, x )? = 0,
(12)
?t
?W = 0.

System (12) admits the in?nite-dimensional Lie algebra with the in?nitesimal opera-
tors
Pa = ?xa , Jab = xa ?xb ? xb ?xa ,
P 0 = ?t ,
i? 1?
Qa = Ua ?xa + Ua xa (??? ? ? ? ??? ) + Ua xa ?W , a = 1, n,
2 2
n
D = xc ?xc + 2t?t ? (??? + ? ? ??? ) ? 2W ?W , (13)
2
i n
A = t2 ?t + txc ?xc + xc xc (??? ? ? ? ??? ) ? t (??? + ? ? ??? ) ? 2W t?W ,
4 2
QB = iB(??? ? ? ??? ) + B?W , Z1 = ??? , Z2 = ? ? ??? ,
? ?

where Ua (t) (a = 1, n ) and B(t) are arbitrary smooth functions. In particular,
algebra (13) includes the Galilei operator (8).
(ii) The condition for the potential is the heat equation.
??
i + ?? + W (t, x )? = 0,
(14)
?t
W0 + ??W = 0.

The maximal invariance algebra of system (14) is

Pa = ?xa , Jab = xa ?xb ? xb ?xa ,
P 0 = ?t ,
n
D = 2t?t + xc ?xc ? (??? + ? ? ??? ) ? 2W ?W ,
2
Z1 = ??? , Z2 = ? ??? , Z3 = it (??? ? ? ? ??? ) + ?W .
?

(iii) The condition for the potential is the wave equation.
??
i + ?? + W (t, x )? = 0,
(15)
?t
2W = 0.

The maximal invariance algebra of system (15) is

Z2 = ? ? ??? ,
Jab = xa ?xb ? xb ?xa ,
P 0 = ?t , Pa = ?xa , Z1 = ??? ,
Z3 = it (??? ? ? ? ??? ) + ?W , Z4 = it2 (??? ? ? ? ??? ) + 2t?W .

(iv) The condition for the potential is the Hamilton–Jacobi equation.
??
i + ?? + W (t, x )? = 0,
?t (16)
?W ?W ?W
?? = 0.
?t ?xa ?xa
The Schr?dinger equation with variable potential
o 281

The maximal invariance algebra is
P0 = ?t , Pa = ?xa , Jab = xa ?xb ? xb ?xa ,
Z1 = ??? , Z2 = ? ? ??? , Z3 = it(??? ? ? ? ??? ) + ?W .

(v) Consider very important and interesting case in (1 + 1)-dimensional space-time
where the condition for the potential is the KdV equation.
?? ? 2 ?
i + + W (t, x)? = 0,
?x2
?t (17)
?3W
?W ?W
+ ?1 W + ?2 = F (|?|), ?1 = 0.
?x3
?t ?x
For an arbitrary F (|?|), system (17) is invariant under the Galilei operator and
the maximal invariance algebra is the following:
P1 = ?x , Z = i (??? ? ? ? ??? ) ,
P 0 = ?t ,
(18)
i 2 1
x + t (??? ? ? ? ??? ) + ?W .
G = t?x +
2 ?1 ?1
For F = C = const, system (17) admits the extension, namely, it is invariant under
the algebra P0 , P1 , G, Z1 , Z2 , where P0 , P1 , G have the form (18) and Z1 = ??? ,
Z2 = ? ? ??? .
The Galilei operator G generates the following transformations:
1
t = t, x = x + ?t, W =W + ?,
?1
i i i
?x + ?t + ?2 t ,
? = ? exp
2 ?1 4
i i i
?? = ? ? exp ? ?x ? ?t ? ?2 t ,
2 ?1 4
where ? is a group parameter. Here, it is important that ?1 = 0, since otherwise,
system (17) does not admit the Galilei operator.

4 Finite-dimensional subalgebras
Algebra (3) is in?nite-dimensional. We select certain ?nite-dimensional subalgebras
from it. In particular, we give the examples of functions Ua (t) and B(t), for which the
subalgebra generated by the operators
(19)
P0 , Pa , Jab , Qa , QB , Z1 , Z2
is ?nite-dimensional.
(a) Ua (t) = exp(?t). In this case, subalgebra (19) has the form
P0 , Pa , Jab , Z1 , Z2 ,
i 1
Qa = e?t ?xa + ?xa (??? ? ? ? ??? ) + ? 2 xa ?W , a = 1, n,
2 2
?
QB = e (i??? ? i? ??? + ??W ) .
?t
282 W.I. Fushchych, Z.I. Symenoh, I.M. Tsyfra

(b) Ua (t) = C1 cos(?t) + C2 sin(?t). Then subalgebra (19) has the form:
P0 , Pa , Jab , Z1 , Z2 ,
i 12
Q(1) = cos(?t)?xa ? ? sin(?t)xa (??? ? ? ? ??? ) ? ? cos(?t)xa ?W ,
a
2 2
i 12
Q(2) = sin(?t)?xa + ? cos(?t)xa (??? ? ? ? ??? ) ? ? sin(?t)xa ?W ,
a
2 2
X1 = i sin(?t) (??? ? ? ? ??? ) + ? cos(?t)?W ,
X2 = i cos(?t) (??? ? ? ? ??? ) ? ? sin(?t)?W .

(c) Ua (t) = C1 tk + C2 tk?1 + · · · + Ck t + Ck+1 . Then subalgebra (19) has the form:
P0 , Pa , Jab , Z1 , Z2 ,
i 1
Q(1) = tk ?xa + ktk?1 xa (??? ? ? ? ??? ) + k(k ? 1)tk?2 xa ?W ,
a
2 2
i 1
Q(2) = tk?1 ?xa + (k ? 1)tk?2 xa (??? ? ? ? ??? ) + (k ? 1)(k ? 2)tk?3 xa ?W ,
a
2 2
·································
i
Q(k) = t?xa + xa (??? ? ? ? ??? ) ,
a
2
QB = it (??? ? ? ? ??? ) + ?W ,
(1)

·································
= it2k?2 (??? ? ? ? ??? ) + (2k ? 2)t2k?3 ?W .
(2k?2)
QB

5 The Schr?dinger equation with convection term
o
Consider equation (1) for W = 0, i.e., the Schr?dinger equation with convection term
o
?? ??
(20)
i + ?? = Va ,
?t ?xa
where ? and Va (a = 1, n ) are complex functions of t and x. For extension of
symmetry, we again regard the functions Va as dependent variables. Note that the
requirement that the functions Va are complex is essential for symmetry of (20).
Let us investigate symmetry properties of (20) in the class of ?rst-order di?erential
operators
X = ? µ ?xµ + ??? + ? ? ??? + ?a ?Va + ??a ?Va ,
?

where ? µ , ?, ? ? , ?a , ??a are functions of t, x, ?, ? ? , Va , Va .
?

Theorem 2. Equation (20) is invariant under the in?nite-dimensional Lie algebra
with the in?nitesimal operators
QA = 2A?t + Axc ?xc ? iAxc ?Vc ? ?Vc? ? A Vc ?Vc + Vc? ?Vc? ,
? ? ?
Qab = Eab xa ?xb ? xb ?xa + Va ?Vb ? Vb ?Va + Va ?Vb? ? Vb? ?Va ?
?
?

?
? iEab xa ?V ? xb ?V ? xa ?V ? + xb ?V ? , (21)
a
b a
b
?
Qa = Ua ?xc ? iUa ?Va ? ?Va ,
?

Z1 = ??? , Z2 = ? ? ??? , Z3 = ?? , Z4 = ??? ,
The Schr?dinger equation with variable potential
o 283

where A, Eab , Ua are arbitrary smooth functions of t. We mean summation over the
index c and no summation over indices a and b.
This theorem is proved by analogy with the previous one.
Note that algebra (21) includes, as a particular case, the Galilei operator of the
form:
Ga = t?xa ? i?Va + i?Va . (22)
?

This operator generates the following ?nite transformations:
t = t, xa = xa + ?a t, xb = xb (b = a),
?? = ?? , ? ?
Va = Va ? i?a ,
? = ?, Va = Va + i?a ,
where ?a is an arbitrary real parameter. Operator (22) is essentially di?erent from
the standard Galilei operator (8) of the Schr?dinger equation, and we cannot derive
o
operator (8) from algebra (21).
Consider now the system of equation (20) with the additional condition for the
potentials Va , namely, the complex Euler equation:
?? ??
i + ?? = Va ,
?t ?xa
(23)
?Va ?Va ??
? Vb
i = F (|?|) .
?t ?xb ?xa
Here, ? and Va are complex dependent variables of t and x, F is an arbitrary functi-
on of |?|. The coe?cients of the second equation of the system provide the broad
symmetry of this system.
Let us investigate the symmetry classi?cation of system (23). Consider the followi-
ng ?ve cases.
1. F is an arbitrary smooth function. The maximal invariance algebra is P0 , Pa ,
Jab , Ga , where
P0 = ?t , Pa = ?xa ,
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