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o 483

PDEs

(1) ?1xb ?2xb = 0,

(2) B1a (?1 )?1xb ?1xb + B2a (?2 )?2xb ?2xb + Ra (t) = 0, a = 1, 2,

(7)

(3) 2?axb Qxb + Q(i?at + ?a ), a = 1, 2,

B01 (?1 )?1xb ?1xb + B02 (?1 )?2xb ?2xb Q + iQt + Q + R0 (t)Q ?

(4)

? V (x1 , x2 )Q = 0.

Thus, to solve the problem of SV for the linear Schr?dinger equation it is necessary

o

to construct general solution of system of nonlinear PDEs (7). Roughly speaking, to

solve a linear equation one has to solve a system of nonlinear equations! This is the

reason why so far there is no complete description of all coordinate systems providing

separability of the four-dimensional wave equation [3].

But in the case involved we have succeeded in integrating system of nonlinear

PDEs (7). Our approach to integration of it is based on the following change of

variables (hodograph transformation)

z0 = t, z1 = Z1 (t, ?1 , ?2 ), z2 = Z2 (t, ?1 , ?2 ), v1 = x1 , v2 = x2 ,

where z0 , z1 , z2 are new independent and v1 , v2 are new dependent variables corres-

pondingly.

Using the hodograph transformation determined above we have constructed the

general solution of Eqs. (1)–(3) from Eq. (7). It is given up to the equivalence relation

(6) by one of the following formulas:

(1) ?1 = A(t)x1 + W1 (t), ?2 = B(t)x2 + W2 (t),

? ? ? ?

i A2 B2 i W1 W2

Q(t, x) = exp ? x1 + x2 ? x1 + x2 ;

4A B 2 A B

1 x1

?1 = ln(x2 + x2 ) + W (t), ?2 = arctan ,

(2) 1 2

2 x2

?

iW 2

Q(t, x) = exp ? (x1 + x2 ) ;

2

4

(8)

1

x1 = W (t)(?1 ? ?2 ) + W1 (t), x2 = W (t)?1 ?2 + W2 (t),

2 2

(3)

2

?

iW i? ?

Q(t, x) = exp (x1 ? W1 )2 + (x2 ? W2 )2 + (W1 x1 + W2 x2 ) ;

4W 2

(4) x1 = W (t) cosh ?1 cos ?2 + W1 (t), x2 = W (t) sinh ?1 sin ?2 + W2 (t),

?

iW i? ?

Q(t, x) = exp (x1 ? W1 )2 + (x2 ? W2 )2 + (W1 x1 + W2 x2 ) ;

4W 2

Here A, B, W , W1 , W2 are arbitrary smooth functions on t.

Substituting the obtained expressions for the functions Q, ?1 , ?2 into the last

equation from the system (7) and splitting with respect to variables x1 , x2 we get

explicit forms of potentials V (x1 , x2 ) and systems of nonlinear ODEs for unknown

functions A(t), B(t), W (t), W1 (t), W2 (t). We have succeeded in integrating these

and in constructing all coordinate systems providing SV in the initial equation (1).

484 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

Here we consider in detail integration of the fourth equation of system (7) for the

case 2 from Eq. (8), since computations needed are not so lengthy as for other cases.

First, we make several important remarks which introduce an equivalence relation

on the set of potentials V (x1 , x2 ).

Remark 1. The Schr?dinger equation with the potential

o

V (x1 , x2 ) = k1 x1 + k2 x2 + k3 + V1 (k2 x1 ? k1 x2 ), (9)

where k1 , k2 , k3 are constants, is transformed to the Schr?dinger equation with the

o

potential

V (x1 , x2 ) = V1 (k2 x1 ? k1 x2 ) (10)

by the following change of variables:

x = x + t2 k,

t = t,

(11)

i2

(k1 + k2 )t3 + it(k1 x1 + k2 x2 ) + ik3 t .

2

u = u exp

3

It is readily seen that the class of Ans?tze (5) is transformed into itself by the

a

above change of variables. That is why, potentials (9) and (10) are considered as

equivalent.

Remark 2. The Schr?dinger equation with the potential

o

x1

(x2 + x2 )?1

V (x1 , x2 ) = k(x2 + x2 ) + V1 (12)

1 2 1 2

x2

with k = const is reduced to the Schr?dinger equation with the potential

o

x1

(x12 + x12 )?1 (13)

V (x1 , x2 ) = V1

x2

by the change of variables

x = ?(t)x, u = u exp i?(t)(x2 + x2 ) + ?(t) ,

t = ?(t), 1 2

where ?(t), ?(t), ?(t), ?(t) is an arbitrary solution of the system of ODEs

? ?

? ? 4? 2 = k, ? ? 4?? = 0, ? ? ? 2 = 0,

? ? ? + 4? = 0

such that ? = 0.

Since the above change of variables does not alter the structure of the Ansatz (5),

when classifying potentials V (x1 , x2 ) providing separability of the corresponding

Schr?dinger equation, we consider potentials (12), (13) as equivalent.

o

Remark 3. It is well-known (see e.g. [11, 12]) that the general form of the invariance

group admitted by Eq. (1) is as follows

t = F (t, ?), xa = ga (t, x, ?), u = h(t, x, ?)u + U (t, x),

a = 1, 2,

where ? = (?1 , ?2 , . . . , ?n ) are group parameters and U (t, x) is an arbitrary solution

of Eq. (1).

Variable separation in the time-dependent Schr?dinger equation

o 485

The above transformations also do not alter the structure of the Ansatz (5). That

is why, systems of coordinates t , x1 , x2 and t, x1 , x2 are considered as equivalent.

Now we turn to the integration of the fourth equation of system (7). Substituting

into it the expressions for the functions ?1 , ?2 , Q given by formulas (2) from Eq. (8)

we get

1? ?

V (x1 , x2 ) = B01 (?1 ) + B02 (?2 ) exp{?2(?1 ? W )} + (W ? W 2 ) ?

4 (14)

?

? exp{2(?1 ? W )} + R0 (t) ? iW .

In the above equality B01 , B02 , R0 (t), W (t) are unknown functions to be determi-

ned from the requirement that the right-hand side of (14) does not depend on t.

Differentiating Eq. (14) with respect to t and taking into account the equalities

?

?1t = W , ?2t = 0

we have

?

? ?

W exp{?2(?1 ? W )}B01 + ?(t) exp{2(?1 ? W )} + ?(t) = 0, (15)

?

? ? ?

where ?(t) = 1 (W ? W 2 ), ?(t) = R0 ? iW .

4

? ?

Cases W = 0 and W = 0 have to be considered separately.

?

Case 1. W = 0. In this case W = C = const, R0 = 0. Since coordinate systems

?1 , ?2 and ?1 + C1 , ?2 + C2 are equivalent with arbitrary constants C1 , C2 , choosing

C1 = ?C, C2 = 0 we can put C = 0. Hence it immediately follows that

1 x1

(x2 + x2 )?1 ,

ln(x2 + x2 ) + B02 arctan

V (x1 , x2 ) = B01 1 2 1 2

2 x2

where B01 , B02 are arbitrary functions. And what is more, the Schr?dinger equa-

o

tion (1) with such potential separates only in one coordinate system

1 x1

ln(x2 + x2 ), (16)

?1 = ?2 = arctan .

1 2

2 x2

? ?

Case 2. W = 0. Dividing Eq. (14) into W exp{?2(?1 ? W )} and differentiating the

equality obtained with respect to t we get

d d ? ? ?1

?(W )?1 exp{?4W } + exp{2?1 }

??

exp{4?1 } ?(W ) exp{?2W } = 0,

dt dt

whence

d d ? ? ?1

?(W )?1 exp{?4W } = 0,

?? ?(W ) exp{?2W } = 0.

dt dt

Integration of the above ODEs yields the following result:

? = C1 exp{4W } + C2 , ? = C3 exp{2W } + C4 ,

where C? , ? = 1, 4 are arbitrary real constants.

Inserting the result obtained into Eq. (15) we get an equation for B01

?

B01 = ?4C1 exp{4?1 } ? 2C3 exp{2?1 },

486 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

which general solution reads

B01 = ?C1 exp{4?1 } ? C3 exp{2?1 } + C5 .

In the above equality C5 is an arbitrary real constant.

Substituting the expressions for ?, ?, B01 into Eq. (14) we have the explicit form

of the potential V (x1 , x2 )

x1

+ C5 (x2 + x2 )?1 + C2 (x2 + x2 ) + C4 ,

V (x1 , x2 ) = B02 arctan 1 2 1 2

x2

where B02 is an arbitrary function.

By force of the Remarks 1, 2 we can choose C2 = C4 = 0. Furthermore, due to

arbitrariness of the function B02 we can put C5 = 0.

?

Thus, the case W = 0 leads to the following potential:

x1

(x2 + x2 )?1 . (17)

V (x1 , x2 ) = B02 arctan 1 2

x2

Substitution of the above expression into Eq. (14) yields second-order nonlinear

ODE for the function W = W (t)

? ?

W ? W 2 = 4C1 exp{4W }, (18)

while the function R0 is given by the formula

?

R0 = iW + C3 exp{2W }.

Integration of ODE (18) is considered in detail in the Appendix A. Its general

solution has the form

under C1 = 0

1 1

W = ? ln (at ? b)2 ? 4C1 + ln a,

2 2

under C1 = 0

W = a ? ln(t + b).

Substituting obtained expressions for W into formulas (2) from (8) and taking into

account the Remark 3 we arrive at the conclusion that the Schr?dinger equation (1)

o

with the potential (17) admits SV in two coordinate systems. One of them is the polar

coordinate system (16) and another one is the following:

1 1 x1

ln(x2 + x2 ) ? ln(t2 ± 1), (19)

?1 = ?2 = arctan .

1 2

2 2 x2

Consequently, the case 2 from Eq. (8) gives rise to two classes of the separable

Schr?dinger equations (1).

o

Cases 1, 3, 4 from Eq. (8) are considered in an analogous way but computations

involved are much more cumbersome. As a result, we obtain the following list of

inequivalent potentials V (x1 , x2 ) providing separability of the Schr?dinger equation.

o

Variable separation in the time-dependent Schr?dinger equation

o 487

(1) V (x1 , x2 ) = V1 (x1 ) + V2 (x2 );

(a) V (x1 , x2 ) = k1 x2 + k2 x?2 + V2 (x2 ), k2 = 0;

1 1

V (x1 , x2 ) = k1 x2 + k2 x2 + k3 x?2 + k4 x?2 ,

(i) k3 k4 = 0,

1 2 1 2

2 2

k1 + k2 = 0, k1 = k2 ;

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

(ii) k1 k2 = 0;

1 1

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