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V (x1 , x2 ) = k1 x?2 + k2 x?2 ;
(iii) 1 2

(b) V (x1 , x2 ) = k1 x2 + V2 (x2 );
1

V (x1 , x2 ) = k1 x2 + k2 x2 + k3 x?2 , k1 k3 = 0, k1 = k2 ;
(i) 1 2 2
2 2
(ii) V (x1 , x2 ) = k1 x1 + k2 x2 , k1 k2 = 0, k1 = k2 ;
V (x1 , x2 ) = k1 x2 + k2 x?2 ,
(iii) k1 = 0;
1 2

(2) V (x1 , x2 ) = V1 (x2 + x2 ) + V2 (x1 /x2 )(x2 + x2 )?1 ;
1 2 1 2
2 ?1
2
(a) V (x1 , x2 ) = V2 (x1 /x2 )(x1 + x2 ) ;
(b) V (x1 , x2 ) = k1 (x2 + x2 )?1/2 , k1 = 0;
1 2
(3) V (x1 , x2 ) = (V1 (?1 ) + V2 (?2 ))(?1 + ?2 )?1 , where ?1 ? ?2 = 2x1 , ?1 ?2 = x2 ;
2 2 2 2

(4) V (x1 , x2 ) = (V1 (?1 ) + V2 (?2 ))(sinh2 ?1 + sin2 ?2 )?1 , where cosh ?1 cos ?2 = x1 ,
sinh ?1 sin ?2 = x2 ;
(5) V (x1 , x2 ) = 0.
In the above formulas V1 , V2 are arbitrary smooth functions, k1 , k2 , k3 , k4 are
arbitrary constants.
It should be emphasized that the above potentials are not inequivalent in a usual
sense. These potentials differ from each other by the fact that the coordinate systems
providing separability of the corresponding Schr?dinger equations are different. As
o
an illustration, we give the Fig. 1, where r = (x2 + x2 )1/2 and by the symbol V (?) ,
1 2
? = 1, 4 we denote the potential given in the above list under the number ?. Down
arrows in the Fig. 1 indicate specifications of the potential V (x1 , x2 ) providing new
possibilities to separate the corresponding Schr?dinger equation (1).
o
The Schr?dinger equation (1) with arbitrary function V (x1 , x2 ) (level 1 of the
o
Fig. 1) admits no separation of variables. Next, Eq. (1) with the “root” potentials
V (?) (level 2), V1 , V2 being arbitrary smooth functions, separates in the Cartesian
(? = 1), polar (? = 2), parabolic (? = 3) and elliptic (? = 4) coordinate systems,
correspondingly. Specifying the functions V1 , V2 (i.e. going down to the lower levels)
new possibilities to separate variables in the Schr?dinger equation (1) arise. For
o
?2
example, Eq. (1) with the potential V2 (x1 /x2 )r , which is a particular case of the
potential V (2) , separates not only in the polar coordinate system (16) but also in
the coordinate systems (19). The Schr?dinger equation with the Coulomb potential
o
?1
k1 r , which is a particular case of the potentials V (2) , V (3) , separates in two coordi-
nate systems (namely, in the polar and parabolic coordinate systems, see below the
Theorem 4). An another characteristic example is a transition from the potential V (1)
to the potential k1 x2 + V2 (x2 ). The Schr?dinger equation with the potential V (1) ad-
o
1
mits SV in the Cartesian coordinate system ?0 = t, ?1 = x1 , ?2 = x2 only, while the
one with the potential k1 x2 + V2 (x2 ) separates in seven (k1 < 0) or in three (k1 > 0)
1
coordinate systems.
488 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

A complete list of coordinate systems providing SV in the Schr?dinger equations
o
with the above given potentials takes two dozen pages. Therefore, we restrict ourself
to considering the Schr?dinger equation with anisotropic harmonic oscillator potential
o
V (x1 , x2 ) = k1 x1 + k2 x2 , k1 = k2 and Coulomb potential V (x1 , x2 ) = k1 (x2 + x2 )?1/2 .
2 2
1 2

3 Separation of variables in the Schr?dinger
o
equation with the anisotropic harmonic
oscillator and the Coulomb potentials
Here we will obtain all coordinate systems providing separability of the Schr?dinger
o
2 2
equation with the potential V (x1 , x2 ) = k1 x1 + k2 x2

iut + ux1 x1 + ux2 x2 = (k1 x2 + k2 x2 )u. (20)
1 2

In the following, we consider the case k1 = k2 , because otherwise Eq. (1) is
reduced to the free Schr?dinger equation (see the Remark 2) which has been studied
o
in detail in [1–3].
Variable separation in the time-dependent Schr?dinger equation
o 489

Explicit forms of the coordinate systems to be found depend essentially on the
signs of the parameters k1 , k2 . We consider in detail the case, when k1 < 0, k2 > 0
(the cases k1 > 0, k2 > 0 and k1 < 0, k2 < 0 are handled in an analogous way). It
means that Eq. (20) can be written in the form
1
iut + ux1 x1 + ux2 x2 + (a2 x2 ? b2 x2 )u = 0, (21)
1 2
4
where a, b are arbitrary non-null real constants (the factor 1 is introduced for further
4
convenience).
As stated above to describe all coordinate systems t, ?1 (t, x), ?2 (t, x) providing
separability of Eq. (20) one has to construct the general solution of system (8) with
V (x1 , x2 ) = ? 1 (a2 x2 ? b2 x2 ). The general solution of Eqs. (1)–(3) from Eq. (7) splits
1 2
4
into four inequivalent classes listed in Eq. (8). Analysis shows that only solutions
belonging to the first class can satisfy the fourth equation of (7).
Substituting the expressions for ?1 , ?2 , Q given by the formulas (1) from (8) into
the equation 4 from (7) with V (x1 , x2 ) = ? 1 (a2 x2 ? b2 x2 ) and splitting with respect
1 2
4
to x1 , x2 one gets
2 2
B01 (?1 ) = ?1 ?1 + ?2 ?1 , B02 (?2 ) = ?1 ?2 + ?2 ?2 ,
· 2
? ?
A A
? ? 4?1 A4 + a2 = 0, (22)
A A
· 2
? ?
B B
? ? 4?1 B 4 ? b2 = 0, (23)
B B
?
?A
?
?1 ? 2?1 ? 2(2?1 ?1 + ?2 )A4 = 0, (24)
A
?
?B
?
?2 ? 2?2 ? 2(2?1 ?2 + ?2 )B 4 = 0. (25)
B
Here ?1 , ?2 , ?1 , ?2 are arbitrary real constants.
Integration of the system of nonlinear ODEs (22)–(25) is carried out in the
Appendix A. Substitution of the formulas (A.2), (A.4)–(A.6), (A.8)–(A.11) into the
corresponding expressions 1 from (8) yields a complete list of coordinate systems
providing separability of the Schr?dinger equation (21). These systems can be trans-
o
formed to canonical form if we use the Remark 3.
The invariance group of Eq. (21) is generated by the following basis operators [11]:
I = u?u , M = iu?u , Q? = U (t, x)?u ,
P 0 = ?t ,
ia
P1 = cosh at?x1 + (x1 sinh at)u?u ,
2
ib
P2 = cos bt?x2 ? (x2 sin bt)u?u , (26)
2
ia
G1 = sinh at?x1 + (x1 cosh at)u?u ,
2
ib
G2 = sin bt?x2 + (x2 cos bt)u?u ,
2
where U (t, x) is an arbitrary solution of Eq. (21).
490 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

Using the finite transformations generated by the infinitesimal operators (26) and
the Remark 3 we can choose in the formulas (A.4)–(A.6), (A.8), (A.10), (A.11)
C3 = C4 = D1 = 0, D3 = D4 = 0, C2 = D2 = 1. As a result, we come to the
following assertion.
Theorem 1. The Schr?dinger equation (21) admits SV in 21 inequivalent coordinate
o
systems of the form

?1 = ?1 (t, x), ?2 = ?2 (t, x), (27)
?0 = t,

where ?1 is given by one of the formulas from the first and ?2 by one of the formulas
from the second column of the Table 1.

Table 1. Coordinate systems proving SV in Eq. (21).

?1 (t, x) ?2 (t, x)
?1 ?2
x2 (sin bt)?1 + ?(sin bt)?2
x1 sinh a(t + C) +? sinh a(t + C)
?1 ?2
x2 (? + sin 2bt)?1/2
x1 cosh a(t + C) +? cosh a(t + C)
x1 exp(±at) + ? exp(±4at) x2
?1/2
x1 ? + sinh 2a(t + C)
?1/2
x1 ? + cosh 2a(t + C)
?1/2
x1 ? + exp(±2at)
x1
Here C, ?, ? are arbitrary real constants.
There is no necessity to consider specially the case when in Eq. (20) k1 > 0,
k2 < 0, since such an equation by the change of independent variables u(t, x1 , x2 ) >
u(t, x2 , x1 ) is reduced to Eq. (21).
Below we adduce without proof the assertions describing coordinate systems provi-
ding SV in Eq. (20) with k1 < 0, k2 < 0 and k1 > 0, k2 > 0.
Theorem 2. The Schr?dinger equation
o
1
iut + ux1 x1 + ux2 x2 + (a2 x2 + b2 x2 )u = 0 (28)
1 2
4
with a2 = 4b2 admits SV in 49 inequivalent coordinate systems of the form (27),
where ?1 is given by one of the formulas from the first and ?2 by one of the formulas
from the second column of the Table 2. Provided a2 = 4b2 one more coordinate
system should be included into the above list, namely

?1 ? ?2 = 2x1 ,
2 2
(29)
?0 = t, ?1 ?2 = x2 .
Variable separation in the time-dependent Schr?dinger equation
o 491

Table 2. Coordinate systems proving SV in Eq. (28).

?1 (t, x) ?2 (t, x)
?1 ?2
x2 (sinh bt)?1 + ?(sinh bt)?2
x1 sinh a(t + C) +? sinh a(t + C)
?1 ?2
x2 (cosh bt)?1 + ?(cosh bt)?2
x1 cosh a(t + C) +? cosh a(t + C)
x1 exp(±at) + ? exp(±4at) x2 exp(±bt) + ? exp(±4bt)
?1/2
x2 (? + sinh 2bt)?1/2
x1 ? + sinh 2a(t + C)
?1/2
x2 (? + cosh 2bt)?1/2
x1 ? + cosh 2a(t + C)
?1/2 ?1/2
x1 ? + exp(±2at) x2 ? + exp(±2bt)
x1 x2
Here C, ?, ? are arbitrary constants.

Table 3. Coordinate systems proving SV in Eq. (30).

?1 (t, x) ?2 (t, x)
?1 ?2
x2 (sin bt)?1 + ?(sin bt)?2
x1 sin a(t + C) +? sin a(t + C)
?1/2
x2 (? + sin 2bt)?1/2
x1 ? + sin 2a(t + C)
x1 x2
Here C, ?, ? are arbitrary constants.
Theorem 3. The Schr?dinger equation
o
1
iut + ux1 x1 + ux2 x2 ? (a2 x2 + b2 x2 )u = 0 (30)
1 2
4
with a2 = 4b2 admits SV in 9 inequivalent coordinate systems of the form (27),
where ?1 is given by one of the formulas from the first and ?2 by one of the
formulas from the second column of the Table 3. Provided a2 = 4b2 , the above list
should be supplemented by the coordinate system (29).
Remark 4. If we consider Eq. (1) as an equation for a complex-valued function u of
three complex variables t, x1 , x2 , then the cases considered in the Theorems 1–3 are
equivalent. Really, replacing, when necessary, a with ia and b by ib we can always
reduce Eqs. (21), (28) to the form (30). It means that coordinate systems presented
in the Tables 1, 2 are complex equivalent to those listed in the Table 3. But if u is
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