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1
Hµ?? = v {(cµ b? ? c? bµ )b? + ?[(dµ a? ? aµ d? )b? ? (d? a? ? a? d? )bµ ]};
?
Gµ = ?dµ , Hµ?? = ?aµ (b? c? ? c? b? ) + a? (bµ c? ? cµ b? );
L9 :
Gµ = aµ , Hµ?? = (bµ c? ? cµ b? )d? ? (b? c? ? c? b? )dµ ;
L10 :
1
Gµ = aµ ? dµ , Hµ?? = [(b? c? ? c? b? )bµ ? (bµ c? ? cµ b? )b? ];
L11 :
2
L12 : Gµ = kµ ,
1
Hµ?? = {(kµ b? ? k? bµ )b? ? ?[(kµ b? ? k? bµ )c? ? (k? b? ? k? b? )cµ ]};
?
Gµ = kµ , Hµ?? = (kµ b? ? k? bµ )c? ? (k? c? ? k? b? )cµ ;
L13 :
1
Gµ = 4bµ , Hµ?? = (bµ k? ? b? kµ )k? ;
L14 :
2
1
Gµ = 4(cµ ? ?bµ ), Hµ?? = (bµ k? ? b? kµ )k? ;
L15 :
2
v
L16 : Gµ = 2cµ ?,
1 1
Hµ?? = ?(aµ d? ? a? dµ )k? ? v k? ? v (bµ c? ? cµ b? )b? ;
? ?
L17 : Gµ = kµ ,
1
{2(b? cµ ? bµ c? )k? + (kµ c? ? k? cµ )b? +
Hµ?? =
1 + ?(? + ?)
Reduction of the self-dual Yang–Mills equations 479

+ (k? bµ ? kµ b? )c? + (? + ?)(kµ b? ? k? bµ )b? +
+ ?(kµ c? ? k? cµ )c? };
Gµ = ?(kµ ? + aµ ? dµ ),
L18 :
Hµ?? = ?[(kµ b? ? k? bµ )b? + (aµ d? ? k? bµ )k? ];
Gµ = cµ , Hµ?? = ?[(kµ b? ? k? bµ )b? + (aµ d? ? a? dµ )k? ];
L19 :
Gµ = cµ + ?kµ , Hµ?? = ?[(aµ d? ? a? dµ )k? + (kµ b? ? k? bµ )b? ];
L20 :
L21 : Gµ = cµ + ??kµ ,
Hµ?? = ?[(aµ d? ? a? dµ )k? + (kµ b? ? k? bµ )b? ? (kµ b? ? k? bµ )k? ];
Gµ = ?(kµ ? + aµ ? dµ ),
L22 :
Hµ?? = ?{(kµ b? ? k? bµ )b? + (kµ c? ? k? cµ )c? +
+ ?[(bµ c? ? cµ bµ )k? ? (b? c? ? c? b? ] + (aµ d? ? a? dµ )k? }.

Here, kµ = aµ + dµ , ? = 1 for ax + dx > 0 and ? = ?1 for ax + dx < 0.
Thus, using symmetry properties of the self-dual Yang–Mills equations and sub-
algebraic structure of the Poincar? algebra, we reduced system of PDE (1) to the
e
system of ordinary differential equations (15). Let us emphasize that system (15)
contains nine equations for twelve functions, which means that it is underdetermined.
This fact simplifies essentially finding its particular solutions.
If one constructs a solution of one of equations (15) (general or particular), then
substitution of the obtained result into the corresponding ansatz from (13). (14) yields
an exact solution of the nonlinear self-dual Yang–Mills equations (1). We intend to
study in detail the reduced system of ordinary differential equations (15) and construct
new classes of exact solutions of equations (1) but this will be a topic of our future
publication.

1. Actor A., Classical solutions of SU(2) Yang–Mills theories, Rev. Mod. Phys., 1979, 51, 3,
461–526.
2. Schwarz F., Symmetry of SU (2) invariant Yang–Mills theories, Lett. Math. Phys., 1982, 6, 5,
355–359.
3. Patera J., Winternitz P., Zassenhaus H., Continuous subgroups of the fundament al groups of
physics. I. General method and the Poincar? group, J. Math. Phys., 1975, 16, 8, 1597–1624.
e
4. .I., .., ..,
, , . , 1991, 304 c.
5. Fushchych W.I., Zhdanov R.Z., Conditional symmetry and reduction of partial differential equations,
Ukr. Math. J., 1992, 44, 7, 970–982.
6. Fushchych W.I., Zhdanov R.Z., Symmetry and exact solutions of nonlinear spinor equations, Phys.
Repts., 1982, 172, 4, 123–174.
7. .., .3., : ,
, , , 1992, 388 c.
8. .., , : , 1978, 400 c.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 480–496.

On the new approach to variable separation
?
in the time-dependent Schrodinger equation
with two space dimensions
R.Z. ZHDANOV, I.V. REVENKO, W.I. FUSHCHYCH
We suggest an effective approach to separation of variables in the Schr?dinger equation
o
with two space variables. Using it we classify inequivalent potentials V (x1 , x2 ) such that
the corresponding Schr?dinger equations admit separation of variables. Besides that,
o
we carry out separation of variables in the Schr?dinger equation with the anisotropic
o
2 2
harmonic oscillator potential V = k1 x1 + k2 x2 and obtain a complete list of coordinate
systems providing its separability. Most of these coordinate systems depend essentially
on the form of the potential and do not provide separation of variables in the free
Schr?dinger equation (V = 0).
o


1 Introduction
The problem of separation of variables (SV) in the two-dimensional Schr?dinger
o
equation

(1)
iut + ux1 x1 + ux2 x2 = V (x1 , x2 )u

as well as the most of classical problems of mathematical physics can be formulated
in a very simple way (but this simplicity does not, of course, imply an existence
of easy way to its solution). To separate variables in Eq. (1) one has to construct
such functions R(t, x), ?1 (t, x), ?2 (t, x) that the Schr?dinger equation (1) after being
o
rewritten in the new variables
z0 = t, z1 = ?1 (t, x), z2 = ?2 (t, x),
(2)
v(z0 , z) = R(t, x)u(t, x)

separates into three ordinary differential equations (ODEs). From this point of view
the problem of SV in Eq. (1) is studied in [1–4].
But no less of an important problem is the one of description of potentials V (x1 , x2 )
such that the Schr?dinger equation admits variable separation. That is why saying
o
about SV in Eq. (1) we imply two mutually connected problems. The first one is to
describe all such functions V (x1 , x2 ) that the corresponding Schr?dinger equation (1)
o
can be separated into three ODEs in some coordinate system of the form (2) (classi-
fication problem). The second problem is to construct for each function V (x1 , x2 )
obtained in this way all coordinate systems (2) enabling us to carry out SV in Eq. (1).
Up to our knowledge, the second problem has been solved provided V = 0 [2,
3] and V = ?x?2 + ?x?2 [1]. The first one was considered in a restricted sense
1 2
in [4]. Authors using symmetry approach to classification problem obtained some
potentials providing separability of Eq. (1) and carried out SV in the corresponding
J. Math. Phys., 1995, 36, 10, P. 5506–5521.
Variable separation in the time-dependent Schr?dinger equation
o 481

Schr?dinger equation. But their results are far from being complete and systematic.
o
The necessary and sufficient conditions imposed on the potential V (x1 , x2 ) by the
requirement that the Schr?dinger equation admits symmetry operators of an arbitrary
o
order are obtained in [5]. But so far there is no systematic and exhaustive description
of potentials V (x1 , x2 ) providing SV in Eq. (1).
To be able to discuss the description of all potentials and all coordinate systems
making it possible to separate the Schr?dinger equation one has to give a definition
o
of SV. One of the possible definitions of SV in partial differential equations (PDEs)
is proposed in our article [6]. It is based on the concept of Ansatz suggested by
Fushchych [7] and on ideas contained in the article by Koornwinder [8]. The said
definition is quite algorithmic in the sense that it contains a regular algorithm of
variable separation in partial differential equations which can be easily adapted to
handle both linear [6, 9] and nonlinear [10] PDEs. In the present article we apply the
said algorithm to solve the problem of SV in Eq. (1).
Consider the following system of ODEs:
d?0
i = U0 (t, ?0 ; ?1 , ?2 ),
dt
(3)
d 2 ?1 d 2 ?2
d?1 d?2
2 = U1 ?1 , ?1 , d? ; ?1 , ?2 , 2 = U2 ?2 , ?2 , d? ; ?1 , ?2 ,
d?1 d?2
1 2

where U0 , U1 , U2 are some smooth functions of the corresponding arguments, ?1 , ?2 ?
R1 are arbitrary parameters (separation constants) and what is more
2 2
?Uµ
(4)
rank =2
??a µ=0 a=1

(the last condition ensures essential dependence of the corresponding solution with
separated variables on ?1 , ?2 , see [8]).
Definition 1. We say that Eq. (1) admits SV in the system of coordinates t, ?1 (t, x),
?2 (t, x) if substitution of the Ansatz

u = Q(t, x)?0 (t)?1 ?1 (t, x) ?2 ?2 (t, x) (5)

into Eq. (1) with subsequent exclusion of the derivatives d?0 /dt, d2 ?1 /d?1 , d2 ?2 /d?2
2 2

according to Eqs. (3) yields an identity with respect to ?0 , ?1 , ?2 , d?1 /d?1 ,
d?2 /d?2 , ?1 , ?2 .
Thus, according to the above definition to separate variables in Eq. (1) one has

(i) to substitute the expression (5) into (1),
d2 ?1
d2 ?2 /d?2 with the help of Eqs. (3),
2
(ii) to exclude derivatives d?0 /dt, 2,
d?1

(iii) to split the obtained equality with respect to the variables ?0 , ?1 , ?2 , d?1 /d?1 ,
d?2 /d?2 , ?1 , ?2 considered as independent.

As a result one gets some over-determined system of PDEs for the functions
Q(t, x), ?1 (t, x), ?2 (t, x). On solving it one obtains a complete description of all
coordinate systems and potentials providing SV in the Schr?dinger equation. Natural-
o
ly, an expression complete description makes sense only within the framework of our
482 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

definition. So if one uses a more general definition it may be possible to construct new
coordinate systems and potentials providing separability of Eq. (1). But all solutions
of the Schr?dinger equation with separated variables known to us fit into the scheme
o
suggested by us and can be obtained in the above described way.



2 Classification of potentials V (x1 , x2 )
We do not adduce in full detail computations needed because they are very cumber-
some. We shall restrict ourselves to pointing out main steps of the realization of the
above suggested algorithm.
First of all we make a remark, which makes life a little bit easier. It is readily
seen that a substitution of the form

Q > Q = Q?1 (?1 )?2 (?2 ),
(6)
?a > ?a = ?a (?a ), a = 1, 2, ?a > ?a = ?a (?1 , ?2 ), a = 1, 2,

does not alter the structure of relations (3), (4), and (5). That is why, we can introduce
the following equivalence relation:

(?1 , ?2 , Q) ? (?1 , ?2 , Q )

provided Eq. (6) holds with some ?a , ?a , ?a .
Substituting Eq. (5) into Eq. (1) and excluding the derivatives d?0 /dt, d2 ?1 /d?1 ,
2

d2 ?2 /d?2 with the use of equations (3) we get
2


i(Qt ?0 ?1 ?2 + QU0 ?1 ?2 + Q?1t ?0 ?1 ?2 + Q?2t ?0 ?1 ?2 ) + ( Q)?0 ?1 ?2 +
? ?
+ 2Qxa ?1xa ?0 ?1 ?2 + 2Qxa ?2xa ?0 ?1 ?2 + Q ( ?1 )?0 ?1 ?2 +
? ? ?
+ ( ?2 )?0 ?1 ?2 + ?1xa ?1xa ?0 U1 ?2 + ?2xa ?2xa ?0 ?1 U2 +
?
+ 2?1xa ?2xa ?0 ?1 ?2 = V Q?0 ?1 ?2 ,
??

where the summation over the repeated index a from 1 to 2 is understood. Hereafter
an overdot means differentiation with respect to a corresponding argument and =
2 2
?x1 + ?x2 .
Splitting the equality obtained with respect to independent variables ?1 , ?2 ,
d?1 /d?1 , d?2 /d?2 , ?1 , ?2 we conclude that ODEs (3) are linear and up to the
equivalence relation (6) can be written in the form

d?0
i = ?1 R1 (t) + ?2 R2 (t) + R0 (t) ?0 ,
dt
d 2 ?1
2 = ?1 B11 (?1 ) + ?2 B12 (?1 ) + B01 (?1 ) ?1 ,
d?1
d 2 ?2
2 = ?1 B21 (?2 ) + ?2 B22 (?2 ) + B02 (?2 ) ?2
d?2

and what is more, functions ?1 , ?2 , Q satisfy an over-determined system of nonlinear

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( 122 .)



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