ñòð. 62 |

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 ,

ñòð. 62 |