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a complex-valued function of real variables t, x1 , x2 it is not the case.
Theorem 4. The Schr?dinger equation with the Coulomb potential
o

iut + ux1 x1 + ux2 x2 ? k1 (x2 + x2 )?1/2 u = 0
1 2

admits SV in two coordinate systems (16), (29).
It is important to note that explicit forms of coordinate systems providing separabi-
lity of Eqs. (21), (28), (30) depend essentially on the parameters a, b contained in
the potential V (x1 , x2 ). It means that the free Schr?dinger equation (V = 0) does not
o
admit SV in such coordinate systems. Consequently, they are essentially new.
492 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

4 Conclusion
In the present paper we have studied the case when the Schr?dinger equation (1)
o
separates into one first-order and two second-order ODEs. It is not difficult to prove
that there are no functions Q(t, x), ?µ (t, x), µ = 0, 1, 2 such that the Ansatz

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

separates Eq. (1) into three second-order ODEs (see Appendix B). Nevertheless, there
exists a possibility for Eq. (1) to be separated into two first-order and one second-order
ODEs or into three first-order ODEs. This is a probable source of new potentials and
new coordinate systems providing separability of the Schr?dinger equation. It should
o
be said that separation of the two-dimensional wave equation

utt ? uxx = V (x)u

into one first-order and one second-order ODEs gives no new potentials as compared
with separation of it into two second-order ODEs. But for some already known
potentials new coordinate system providing separability of the above equation are
obtained [9].
Let us briefly analyze a connection between separability of Eq. (1) and its symmet-
ry properties. It is well-known that each solution of the free Schr?dinger equation with
o
separated variables is a common eigenfunction of two mutually commuting second-
order symmetry operators of the said equation [2, 3]. And what is more, separation
constants ?1 , ?2 are eigenvalues of these symmetry operators.
We will establish that the same assertion holds for the Schr?dinger equation (1).
o
Let us make in Eq. (1) the following change of variables:

u = Q(t, x)U t, ?1 (t, x), ?2 (t, x) , (31)

where (Q, ?1 , ?2 ) is an arbitrary solution of the system of PDEs (7).
Substituting the expression (31) into (1) and taking into account equations (7) we
get

Q iUt + U?1 ?1 ? B01 (?1 )U ?1xa ?1xa + U?2 ?2 ? B02 (?2 )U ?2xa ?2xa = 0.(32)

Resolving Eqs. (2) from the system (7) with respect to ?1xa ?1xa and ?2xa ?2xa we
have
1
R2 (t)B21 (?2 ) ? R1 (t)B22 (?2 ) ,
?1xa ?1xa =
?
1
= R1 (t)B12 (?1 ) ? R2 (t)B11 (?1 ) ,
?2xa ?2xa
?
where ? = B11 (?1 )B22 (?2 ) ? B12 (?1 )B21 (?2 ) (? = 0 by force of the condition (4)).
Substitution of the above equalities into Eq. (32) with subsequent division by
Q = 0 yields the following PDE:
R1 (t)
B12 (?1 ) U?2 ?2 ? B02 (?2 )U ? B22 (?2 ) U?1 ?1 ? B01 (?1 )U +
iUt +
? (33)
R2 (t)
B21 (?2 ) U?1 ?1 ? B01 (?1 )U ? B11 (?1 ) U?2 ?2 ? B02 (?2 )U = 0.
+
?
Variable separation in the time-dependent Schr?dinger equation
o 493

Thus, in the new coordinates t, ?1 , ?2 , U (t, ?1 , ?2 ) Eq. (1) takes the form (33).
By direct (and very cumbersome) computation one can check that the following
second-order differential operators:
B22 (?2 ) 2 B12 (?1 ) 2
??1 ? B01 (?1 ) ? ??2 ? B02 (?2 ) ,
X1 =
? ?
B21 (?2 ) 2 B11 (?1 ) 2
X2 = ? ??1 ? B01 (?1 ) + ??2 ? B02 (?2 ) ,
? ?
commute under arbitrary B0a , Bab , a, b = 1, 2, i.e.
[X1 , X2 ] ? X1 X2 ? X2 X1 = 0. (34)
After being rewritten in terms of the operators X1 , X2 Eq. (33) reads
i?t ? R1 (t)X1 ? R2 (t)X2 U = 0.
Since the relations
i?t ? R1 (t)X1 ? R2 (t)X2 , Xa = 0, (35)
a = 1, 2
hold, operators X1 , X2 are mutually commuting symmetry operators of Eq. (33).
Furthermore, solution of Eq. (33) with separated variables U = ?0 (t)?1 (?1 )?2 (?2 )
satisfies the identities
(36)
Xa U = ?a U, a = 1, 2.
Consequently, if we designate by X1 , X2 the operators X1 , X2 written in the
initial variables t, x, u, then we get from (34)–(36) the following equalities:
i?t + ? V (x1 , x2 ), Xa = 0, a = 1, 2,
X1 , X2 = 0, Xa u = ?a u, a = 1, 2.
where u = Q(t, x)?0 (t)?1 (?1 )?2 (?2 ).
It means that each solution with separated variables is a common eigenfunction of
two mutually commuting symmetry operators X1 , X2 of the Schr?dinger equation (1),
o
separation constants ?1 , ?2 being their eigenvalues.
Detailed study of the said operators as well as analysis of separated ODEs for
functions ?µ , µ = 0, 2 (in the way as it is done for the free Schr?dinger equation in
o
[2, 3]) is in progress and will be a topic of our future publications.
Acknowledgments. When the paper was in the last stage of preparation, one of
the authors (R. Zhdanov) was supported by the Alexander von Humboldt Founda-
tion. Taking an opportunity he wants to express his gratitude to Director of Arnold
Sommerfeld Institute for Mathematical Physics Professor H.-D. Doebner for hospi-
tality.


Appendix A. Integration of nonlinear ODEs (22)–(25)
Evidently, equations (22)–(25) can be rewritten in the following unified form:
· 2
y
? y
? y
?
? ? 4?y 4 = k, z ? 2z ? 2(2?z + ?)y 4 = 0. (A1)
? ?
y y y
494 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

Provided k = ?a2 < 0, system (A.1) coincides with Eqs. (22), (24) and under
k = b2 > 0 – with Eqs. (23), (25).
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 have 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 can choose in (A.1) ? = 0.
The case 1. ? = 0. On making in (A.1) the change of variables
(A2)
w = y/y,
? v = z/y
we get
w = w2 + k, v + kv = 2?y 3 . (A3)
? ?
First, we consider the case k = ?a2 < 0. Then the general solution of the first
equation from (A.3) is given by one of the formulas
w = ?a coth a(t + C1 ), w = ?a tanh a(t + C1 ), w = ±a, C1 ? R1 ,
whence
y = C2 sinh?1 a(t + C1 ), y = C2 cosh?1 a(t + C1 ),
(A4)
y = C2 exp(±at), C2 ? R1 .
The second equation of system (A.3) is a linear inhomogeneous ODE. Its general
solution after being substituted into (A.2) yields the following expression for z(t):
4
?C2
(C3 cosh at + C4 sinh at) sinh?1 a(t + C1 ) + sinh?2 a(t + C1 ),
2
a
4
?C2
(C3 cosh at + C4 sinh at) cosh a(t + C1 ) + 2 cosh?2 a(t + C1 ),
?1 (A5)
a
4
?C2
exp(±4at), C3 , C4 ? R1 .
(C3 cosh at + C4 sinh at) exp(±at) + 2
4a
The case k = b2 > 0 is treated in an analogous way, the general solution of (A.1)
being given by the formulas
y = D2 sin?1 b(t + D1 ),
(A6)
4
?D2
?1
b(t + D1 ) + 2 sin?2 b(t + D1 ),
z = (D3 cos bt + D4 sin bt) sin
b
where D1 , D2 , D3 , D4 are arbitrary real constants.
The case 2. ? = 0, ? = 0. On making in Eq. (A.1) the change of variables
y = exp w, v = z/y
we have
w ? w2 = k + ? exp 4w, (A7)
? ? v + kv = 0.
?
Variable separation in the time-dependent Schr?dinger equation
o 495

The first ODE from Eq. (A.7) is reduced to the first-order linear ODE
1 dp(w)
? p(w) = k + ? exp 4w
2 dw
by the substitution w = (p(w))1/2 , whence
?

p(w) = ? exp 4w + ? exp 2w ? k, ? ? R1 .

Equation w = (p(w))1/2 has a singular solution w = C = const such that p(C) = 0.
?
If w = 0, then integrating the equation w = p(w) and returning to the initial variable
? ?
y we get
y(t)
d?
= t + C1 .
? (?? 4 + ?? 2 ? k)1/2
Taking the integral in the left-hand side of the above equality we obtain the general
solution of the first ODE from Eq. (A.1). It is given by the following formulas:
under k = ?a2 < 0
?1/2
y = C2 ? + sinh 2a(t + C1 ) ,
?1/2
(A8)
y = C2 ? + cosh 2a(t + C1 ) ,
?1/2
y = C2 ? + exp(±2at) ,

under k = b2 > 0
?1/2
(A9)
y = D2 ? + sin 2b(t + D1 ) .

Here C1 , C2 , D1 , D2 are arbitrary real constants.
Integrating the second ODE from Eq. (A.7) and returning to the initial variable z
we have
under k = ?a2 < 0

(A10)
z = y(t)(C3 cosh at + C4 sinh at)

under k = b2 > 0

(A11)
z = y(t)(D3 cos bt + D4 sin bt),

where C3 , C4 , D3 , D4 are arbitrary real constants.
Thus, we have constructed the general solution of the system of nonlinear ODEs
(A.1) which is given by the formulas (A.5)–(A.11).


Appendix B. Separation of Eq. (1)
into three second-order ODEs
Suppose that there exists an Ansatz

u = Q(t, x)?0 (?0 (t, x))?1 ?1 (t, x) ?2 ?2 (t, x) (A12)
496 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

which separates the Schr?dinger equation into three second-order ODEs
o
d 2 ?0 d 2 ?1
d?0 d?1
= U0 ?0 , ?0 , ; ? 1 , ?2 , = U1 ?1 , ?1 , ; ? 1 , ?2 ,
2 2
d?0 d?0 d?1 d?1
(A13)
d 2 ?2 d?2
2 = U2 ?2 , ?2 , d? ; ?1 , ?2
d?2 2

according to the Definition 1.
Substituting the Ansatz (A.12) into Eq. (1) and excluding the second derivatives
2 2
d ?µ /d?µ , µ = 0, 2 according to Eqs. (A.13) we get

i(Qt ?0 ?1 ?2 + Q?0t ?0 ?1 ?2 + Q?1t ?0 ?1 ?2 + Q?2t ?0 ?1 ?2 ) + ( Q)?0 ?1 ?2 +
? ? ?

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