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2 2
m2
2 2
? = x0 ?0 + x0 (xa ?a + u ?u2 ) + x ?u2
2
Лiївська та умовна симетрiя системи рiвнянь Гамiльтона–Якобi 275

та нескiнченним оператором R = K(x0 )?u2 , де K(x0 ) — довiльна гладка фун-
кцiя.
Теорема 6. Максимальною лiївською алгеброю iнварiантностi системи рiвнянь
(4), (5) е нескiнченно вимiрна алгебра з iнфiнiтезимальним оператором

X = ??0 + (? + ?)xa + cab xb + ca,n+1 u2 + da ?a +
?
1?2 ? ?
(? ?)(x ? (u2 )2 ) + da xa ? dn+1 u2
+ ?u1 + m ? + ? ?u1 +
4
1
+ (? + ?)u2 + can+1 xa + dn+1 ?u2 ,
?
2
де ?(x0 ), ?(x0 ), da (x0 ), dn+1 (x0 ), ?(x0 ) — довiльнi гладкi функцiї, ca,n+1 = ?cn+1,a
сталi.
3 наведених результатiв випливає, що природним узагальненням рiвняння
Гамiльтона–Якобi є система (3), (4), (5) для двох функцiй u1 i u2 . Внаслiдок
широких симетрiйних властивостей вона є претендентом для опису реальних фi-
зичних процесiв.

1. Boyer С.P., Pena?el M.N., Conformal symmetry of the Hamilton–Jacobi equation and quan-
tization, Nuovo Cim. B, 1976, 31, № 2, 195–210.
2. Серова М.М., О нелинейных уравнениях теплопроводности, инвариантных относительно
группы Галилея, в сб. Теоретико-групповые исследования уравнений математической
физики, Киев, Ин-т математики АН Украины, 1985, 119–123.
3. Fushchych W., Shtelen W., Serov N., Symmetry analysis and exact solutions of equations of
nonlinear mathematical physics, Dordrech, Kluwer Acadamic Publishers, 1993, 436 p.
4. Fushchych W., Chemiha R., Galilei-invariant nonlinear systems of evolution equations, J.
Phys. A: Math. Gen., 1995, 28, 5569–5579.
5. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., Наука, 1978,
400 с.
W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 276–285.


The Schr?dinger equation
o
with variable potential
W.I. FUSHCHYCH, Z.I. SYMENOH, I.M. TSYFRA

We study symmetry properties of the Schr?dinger equation with the potential as
o
a new dependent variable, i.e., the transformations which do not change the form of
the class of equations. We also consider systems of the Schr?dinger equations with
o
certain conditions on the potential. In addition we investigate symmetry properties of
the equation with convection term. The contact transformations of the Schr?dinger
o
equation with potential are obtained.


1 Introduction
Let us consider the following generalization of the Schr?dinger equation
o
?? ??
+ ?? + W (t, x, |?|)? + Va (t, x ) (1)
i = 0,
?t ?xa

?2
where ? = , a = 1, n, ? = ?(t, x ) is an unknown complex function, W =
?xa ?xa
W (t, x, |?|) and Va = Va (t, x ) are potentials of interaction.
When Va = 0 in (1), the standard Schr?dinger equation is obtained. Symmetry
o
properties of this equation were thoroughly investigated (see, e.g., [1–4]). For arbitrary
W (t, x ), equation (1) admits only the trivial group of identical transformations x >
x = x, t > t = t, ? > ? = ? [1, 3].
In [5–7], a method for extending the symmetry group of equation (1) was suggested.
The idea lies in the fact that, in equation (1), we assume that W (t, x, |?|) is a new
dependent variable on equal conditions with ?. This means that equation (1) is
regarded as a nonlinear equation even in the case where the potential W does not
depend on ?. Indeed, equation (1) is a set of equations when V is a certain set of
arbitrary smooth functions.


2 2. Symmetry of the Schr?dinger equation
o
with potential
Using this idea, we obtain the invariance algebra of the Schr?dinger equation with
o
potential, i.e.,
??
+ ?? + W (t, x, |?|)? = 0. (2)
i
?t
J. Nonlinear Math. Phys., 1998, 5, № 1, P. 13–22; Preprint, Department of Mathematics, Li-
nk?ping University, Sweden, 1996, 15 p.
o
The Schr?dinger equation with variable potential
o 277

Theorem 1. Equation (2) is invariant under the in?nite-dimensional Lie algebra
with in?nitesimal operators of the form
Jab = xa ?xb ? xb ?xa ,
i? 1?
Qa = Ua ?xa + Ua xa (??? ? ? ? ??? ) + Ua xa ?W ,
2 2
i?
QA = 2A?t + Axc ?xc + Axc xc (??? ? ? ? ??? ) ?
? (3)
4
?
nA 1 ...
(??? + ? ? ??? ) + ?
? A xc xc ? 2W A ?W ,
2 4
QB = iB (??? ? ? ? ??? ) + B?W , Z1 = ??? , Z2 = ? ? ??? ,
?

where Ua (t), A(t), B(t) are arbitrary smooth functions of t, over the index c we
mean summation from 1 to n, a, b = 1, n, and over the repeated index a there is no
summation. The upper dot stands for the derivative with respect to time.
Note that the invariance algebra (3) includes the operators of space (Ua = 1) and
time (A = 1/2) translations, the Galilei operator (Ua = t), the dilation (A = t) and
projective (A = t2 /2) operators.
Proof of Theorem 1. We seek the symmetry operators of equation (2) in the class
of ?rst-order di?erential operators of the form:
X = ? µ (t, x, ?, ? ? )?xµ + ?(t, x, ?, ? ? )?? +
(4)
+ ? ? (t, x, ?, ? ? )??? + ?(t, x, ?, ? ? , W )?W .
Using the invariance condition [1, 8, 9] of equation (2) under operator (4) and the
fact that W = W (t, x, |?|), i.e., ? ?W = ? ? ??? , we obtain the system of determining
?W
??
equations:
j j 0 a b a b 0 a
?? = ??? = 0, ?a = 0, ?a = ?b , ?b + ?a = 0, ?0 = 2?a ,
a
??? = 0, ??? = 0, ??a = (i/2)?0 ,
? ? ?
?? = 0, ??? ?? = 0, ??? a = ?(i/2)?0 ,
a
(5)
i?0 + ?cc ? ?? W ? + 2W ?n ? + W ? + ?? = 0,
n

?i?0 + ?cc ? ??? W ? ? + 2W ?n ? ? + W ? ? + ?? ? = 0,
? ? ? n

?? = ??? = 0,

where an index j varies from 0 to n, a, b = 1, n, over the repeated index c we mean
the summation from 1 to n, and over the indices a, b there is no summation.
We solve system (5) and obtain the following result:
?
? 0 = 2A, ? a = Axa + C ab xb + Ua , a = 1, n,
i 1? i 1?
Axc xc + Uc xc + B ?, ? ? = ? Axc xc + Uc xc + E ? ? ,
? ?
?=
22 22
1 1 ... n?
? ? ?
A xc xc + Uc xc + B ? iA ? 2W A,
?=
22 2
?
where A, Ua , B are arbitrary functions of t, E = B ? 2inA + C1 , C ab = ?C ba and
C1 are arbitrary constants. The theorem is proved.
278 W.I. Fushchych, Z.I. Symenoh, I.M. Tsyfra

The operators QB generate the ?nite transformations:
t = t, x = x,
? ? = ? ? exp(?iB(t)?), (6)
? = ? exp(iB(t)?),
?
W = W + B(t)?,
where ? is a group parameter, B(t) is an arbitrary smooth function.
Using the Lie equations, we obtain that the following transformations correspond
to the operators Qa :
t = t, xa = Ua (t)?a + xa , xb = xb (b = a),
i? i?
2
? = ? exp Ua Ua ?a + Ua xa ?a ,
4 2
(7)
i? i?
? ? = ? ? exp ? Ua Ua ?a ? Ua xa ?a ,
2
4 2
1? 1? 2
W = W + Ua xa ?a + Ua Ua ?a ,
2 4
where ?a (a = 1, n ) are group parameters, Ua = Ua (t) are arbitrary smooth functions,
there is no summation over the index a. In particular, if Ua (t) = t, then the opera-
tors Qa are the standard Galilei operators
i
Ga = t?xa + xa (??? ? ? ? ??? ) . (8)
2
For the operators QA , it is di?cult to write out the ?nite transformations in the
general form. We consider several particular cases:
(a) A(t) = t. Then
n
QA = 2t?t + xc ?xc ? (??? + ? ? ??? ) ? 2W ?W
2
is a dilation operator generating the transformations
t = t exp(2?), xc = xc exp(?),
n n
? = exp ? ? ?, ? ? = exp ? ? ? ? , (9)
2 2
W = W exp(?2?),
where ? is a group parameter.
(b) A(t) = t2 /2. Then
i n
QA = t2 ?t + txc ?xc + xc xc (??? ? ? ? ??? ) ? t (??? + ? ? ??? ) ? 2tW ?W
4 2
is the operator of projective transformations:
t xc
t= , xc = ,
1 ? µt 1 ? µt
ixc xc µ
? = ?(1 ? µt)n/2 exp , (10)
4(1 ? µt)
?ixc xc µ
? ? = ? ? (1 ? µt)n/2 exp , W = W (1 ? µt)2 ,
4(1 ? µt)
µ is an arbitrary parameter.
The Schr?dinger equation with variable potential
o 279

Consider the example. Let

1 1
(11)
W= 2=xx .
x cc

We describe how new potentials are generated from potential (11) under transforma-
tions (6), (7), (9), (10).
(i) QB :

1 1 1
>W = + B(t)? > W = + B(t)(? + ? ) > · · · ,
W=
xc xc xc xc xc xc

where B(t) is an arbitrary smooth function, ? and ? are arbitrary real parameters.
(ii) Qa :

1
>W ,
W=
xc xc
1 1? 1?
+ Ua Ua ?a + Ua ?a (xa ? Ua ?a ),
2
W=
(xa ? Ua (t)?a )2 + xb xb 4 2
W >W ,
1 1?
+ Ua Ua (?a + ?a2 )+
2
W=
(xa ? Ua (t)(?a + ?a ))2+x x 4
bb
1? 1?
+ Ua (?a + ?a )(xa ? Ua (?a + ?a )) + Ua Ua ?a ?a > · · · ,
2 2
where Ua are arbitrary smooth functions, ?a and ?a are real parameters, there is no
summation over a but there is summation over b (b = a). In particular, if Ua (t) = t,
then we have the standard Galilei operator (8) and

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