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1 x1
ln(x2 + x2 ) + W (t), ?2 = arctg ,
2) ?1 = 1 2
2 x2
?
IW 2
Q(t, x) = exp ? (x1 + x2 ) ,
2
4
1
x1 = W (t)(?1 ? ?2 ) + W1 (t), x2 = W (t)?1 ?2 + W2 (t),
2 2
3)
2 (6)
?
iW i? ?
[(x1 ? W1 )2 + (x2 ? W2 )2 ] + (W1 x1 + W2 x2 ) ,
Q(t, x) = exp
4W 2
4) x1 = W (t) ch ?1 cos ?2 + W1 (t), x2 = W (t) sh ?1 sin ?2 + W2 (t),
?
iW i? ?
[(x1 ? W1 )2 + (x2 ? W2 )2 ] + (W1 x1 + W2 x2 ) .
Q(t, x) = exp
4W 2

Here A, B, W , W1 , W2 are arbitrary smooth functions on t. Dot means differentiation
with respect to t.
Substituting obtained expressions for Q, ?1 , ?2 into the last equation from the
system (5) and splitting with respect to the variables x1 , x2 we get explicit forms of
potentials V (x1 , x2 ) and systems of nonlinear ODE for functions A(t), B(t), W (t),
W1 (t), W2 (t). We have suceeded in integrating these and in constructing all coordinate
systems providing SV in the initial equation (1). Complete list of these systems takes
two dozens of pages, so we are to restrict ourselves to adducing explicit forms of
potentials V (x1 , x2 ) such that the Schr?dinger equation (1) admits SV.
o
x1
(x2 + x2 )?1 ;
V (x) = V1 (x2 + x2 ) + V2
1) 1 2 1 2
x2
x1
(x2 + x2 )?1 ;
2) V (x) = V2 1 2
x2
V (x) = [V1 (?1 ) + V2 (?2 )](?1 + ?2 )?1 ,
2 2
3)
12
where x1 = (?1 ? ?2 ), x2 = ?1 ?2 ;
2
2
V (x) = [V1 (?1 ) + V2 (?2 )](sh2 ?1 + sin2 ?2 )?1 ,
4)
where x1 = ch ?1 cos ?2 , x2 = sh ?1 sin ?2 ;
5) V (x) = V1 (x1 ) + V2 (x2 );
(7)
V (x) = kx2 + V2 (x2 );
6) 1
V (x) = k1 x2 + k2 x?2 + V2 (x2 ), k2 = 0;
7) 1 1
2
8) V (x) = kx1 , k = 0;
V (x) = k1 x2 + k2 x2 , k1 k2 = 0;
9) 1 2
V (x) = k1 x1 + k2 x?2 , k1 k2 = 0;
2
10) 1
V (x) = k1 x1 + k2 x2 + k3 x?2 , k1 k3 = 0;
2
11) 2 2
V (x) = k1 x1 + k2 x2 + k3 x?2 + k4 x?2 , k3 k4 = 0,
2 2 2 2
12) k1 + k2 = 0;
1 2
?2 ?2
13) V (x) = k1 x1 + k2 x2 ;
14) V (x) = 0.
In the above formulae V1 , V2 are arbitrary smooth functions, k, k1 , k2 , k3 , k4 are
arbitrary real constants.
286 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

Note 1. The Schr?dinger equation with the potential
o
x1
(x2 + x2 )?1 ,
V (x) = k(x2 + x2 ) + V1 (8)
k = const,
1 2 1 2
x2
is reduced to the Schr?dinger equation with the potential
o
x1
(x12 + x 2 )?1 (9)
V (x ) = V1
x2
by the change of variables
u = exp(i?(t)x2 + i?(t)).
t = ?(t), x = ?(t)x,
(explicit form of the functions ?(t), ?(t), ?(t), ?(t) depends on the sign of the
parameter k in (8)). Since the above change of variables does not alter the structure
of ansatz (2), when classifying potentials V (x1 , x2 ) providing separability of Eq. (1)
we consider potentials (7), (8) as equivalent.
Note 2. It is well-known (see, e.g. [5, 6]) that the general form of the invariance
group admitted by Eq. (1) is as follows:
(10)
t = f (t, ?), xa = ga (t, x, ?), a = 1, 2, u = h(t, x, ?)u,
where ? = (?1 , ?2 , . . . , ?n ) are group parameters.
Since transformations (10) do not alter the structure of the ansatz (2), systems of
coordinates t , x1 , x2 and t, x1 , x2 are considered as equivalent.
Thus, there exist fourteen inequivalent types of the Schr?dinger equations of the
o
form (1) admitting SV. Consequently, the classification problem for Eq. (1) is solved.
Next, we shall obtain all coordinate systems providing separability of the Schr?-o
2 2
dinger equation having the potential V = k1 x1 + k2 x2 (the harmonic oscillator type
equation). Explicit forms of the coordinate systems to be found depend essentially
on the signs of the parameters k1 , k2 . Here we consider the case, when k1 < 0,
k2 > 0 (the cases k1 > 0, k2 > 0 and k1 < 0, k2 < 0 will be considered in a separate
publication). It means that Eq. (1) can be written in the form
1
iut + ux1 x1 + ux2 x2 + (a2 x2 ? b2 x2 )u = 0, (11)
1 2
4
where a, b, are arbitrary real constants (the factor 1/4 is introduced for further
convenience).
We have proved above that to describe all coordinate systems t, ?1 , ?2 providing
separability of Eq. (11) one has to construct the general solution of system (5). The
general solution of equations 1–3 from (5) splits into four inequivalent classes listed
in (6).
Analysis shows that only solutions belonging to the first class can satisfy equati-
on 4 from (5). Substituting corresponding formulae for ?1 , ?2 , Q into equation 4
from (5) with V = 1 (a2 x2 ? b2 x2 ) and splitting with respect to x1 , x2 one gets
1 2
4
2 2
B01 (?1 ) = ?1 ?1 + ?2 ?1 , B02 (?2 ) = ?1 ?2 + ?2 ?2 ,
· 2
? ?
A A
? ? 4?1 A4 ? a2 = 0, (12a)
A A
Variable separation in the two-dimensional Schr?dinger equation
o 287

· 2
? ?
B B
? ? 4?1 B 4 + b2 = 0, (12b)
B B

?
?1 ? 2 A ?1 ? 2(2?1 ?1 + ?2 )A4 = 0,
? (12c)
?
A
?
A?
?
?2 ? 2 ?2 ? 2(2?1 ?2 + ?2 )B 4 = 0, (12d)
A
Here ?1 , ?2 , ?1 , ?2 are arbitrary real constants.
Integration of the system of nonlinear ODE (12a–d) is carried out in the Appendix.
Substitution of the formulae (A.4)–(A.9) into expressions 1 from (5) yields the comp-
lete list of coordinate systems providing separability of the Schr?dinger equation (11).
o
These systems can be reduced to the canonical form if we use the Note 2. The
invariance group of Eq. (11) is generated by the following basis operators [6]:
ia
P 0 = ?t , I = u?u , M = iu?u , P1 = ch at?x1 + (x1 sh at)u?u ,
2
ib ia
P2 = cos bt?x2 ? (13)
(x2 sin bt)u?u , G1 = sh at?x1 + (x1 ch at)u?u ,
2 2
ib
G2 = sin bt?x1 + (x2 cos bt)u?u .
2
Using the finite transformations generated by the infinitesimal operators (13) and
the Note 2 we may choose in the formulae (A.4)–(A.9) C3 = C4 = D1 = 0, C2 =
D2 = 1, D3 = D4 = 0. As a result we come to the following assertion.
Theorem. The Schr?dinger equation (11) admits SV in 21 inequivalent coordinate
o
systems of the form

(14)
?0 = t, ?1 = ?1 (t, x), ?2 = ?2 (t, x),

where ?1 is given by one of the following formulae:
x1 (sh a(t + C))?1 + a(sh a(t + C))?2 , x1 (ch a(t + C))?1 + a(ch a(t + C))?2 ,
x1 (a + sh 2a(t + C))?1/2 , (15)
x1 exp(±a(t + C)) + a exp(±4a(t + C)),
x1 (a + ch 2a(t + C))?1/2 , x1 (a + exp(±2a(t + C)))?1/2 , x1
and ?2 is given by one of the following formulae:
x2 (sin bt)?1 + ?(sin bt)?2 , x2 (? + sin 2bt)?1/2 , (16)
x2 .

In the above formulae C, ?, ? are arbitrary real parameters.
It is important to note that explicit form of the coordinate systems providing
separability of Eq. (11) depends essentially on the parameters a, b contained in the
potential V (x1 , x2 ). It means that in the free case (V = 0) the Schr?dinger equation
o
does not admit SV in such coordinate systems. Consequently, they are essentially
new.
288 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

Appendix. Integration of nonlinear ODE (12a–d).
Evidently, equations (12a–d) can be rewritten in the following unified form:
· 2
y
? y
? y?
? ? ?y 4 = k, z ? z ? (?z + ?)y 4 = 0. (A.1)
? ?
y y y

Provided k = ?a2 < 0, system (A.1) coincides with equations (12a,c) and under
k = b2 > 0 — with equations (12b,d).
First of all, we note that the function z = z(t) is determined up to addition of an
arbitrary constant. Really, the coordinate functions ?a has the following structure:

?a = yxa + z, a = 1, 2.

But the coordinate system t, ?1 , ?2 is equivalent to the coordinate system t,
?1 + C1 , ?2 + C2 , Ca ? R1 . Hence, it follows that the function z(t) is equivalent to
the function z(t) + C with arbitrary real constant C. Consequently, provided ? = 0,
we may choose in (A.1) ? = 0.
The case 1. a = 0. On making the change of variables in (A.1)
y
? z
(A.2)
w= , v=
y y
we get

w = w2 + k, v + kv = ?y 3 . (A.3)
? ?

First, we consider the case k = ?a2 < 0. Then the general solutions of the first
equation from (A.3) is given by the formulae w = ?a cth a(t + C1 ), w = ?a th a(t +
C1 ), w = ±a, C1 ? R1 , whence

y = C2 sh?1 a(t + C1 ), y = C2 ch?1 a(t + C1 ),
(A.4)
C2 ? R1 .
y = exp[±a(t + C1 )],

The second equation of system (A.3) is linear inhomogeneous ODE. Its general solu-
tion after being substituted into (A.2) yields:
4
?C2 ?2
?1
z = (C3 ch at + C4 sh at) sh a(t + C1 ) + sh a(t + C1 ),
2a2
4
?C2 ?2
z = (C2 ch at + C4 sh at) ch?1 a(t + C1 ) + ch a(t + C1 ),
(A.5)
2a2
?
z = (C3 ch at + C4 sh at) exp[±a(t + C1 )] + 2 exp[±4a(t + C1 )],
8a
C3 , C4 ? R .
1

The case k = b2 > 0 is treated in the analogous way, the general solution of (A.3)
being given by the formulae

y = D2 sin?1 b(t + D1 ),
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