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6 12k1 4k1 6k1 24k1

(3.11d)

2 2 4

k2 k3 k2 k4 k2 k3 k2

, k1 = ?4 , k5 ? ?

C3 = + + 2 = ?5

2k1 12 2k1 3k1 16k1

for cases (3.10a)–(3.10d) correspondingly, we reduce (3.6) to one of the forms (3.8a)–

(3.8d) respectively.

From (2.2), (2.8), (3.4), (3.9)–(3.11) we ?nd the corresponding symmetry operators

3 1 i

Q = p3 + {U, p} ? 2pH + U p + U , (3.12a)

4 2 4

3

Q = p3 + {U, p} ? ?2 t, (3.12b)

4

3 1

Q = p3 + {U, p} + ?3 tH ? {x, p} , (3.12c)

4 4

1 i 1

Q± = v p3 ± ?{{x, p}, p} + {3? ? ? 2 x2 , p}±

4 4

24

(3.12d)

v

?2 3

i

± ? ? + 2x? ? exp(±i?t), ? = ??4 ,

x

2 3

where U and ? are solutions of (3.2) and H is the related Hamiltonian (2.1).

Thus, the Schr?dinger equation (2.1) admits a third-order SO if potential U satis-

o

?es one of the equations (3.8). The explicit form of the corresponding SOs is present

in (3.12).

Higher symmetries and exact solutions of Schr?dinger equation

o 147

4 Algebraic properties of SOs

Let us investigate algebraic properties of SOs de?ned by relations (3.12). We shall see

that these properties are predetermined by the type of equations (3.8) satis?ed by U .

By direct calculations, using (2.3), (2.1) and (3.12), we ?nd the following relations

(4.1a)

[Q, H] = 0,

3 C

Q2 = 8H 2 ? ?1 H ? (4.1)

2 8

if the potential satis?es equation (3.8a) (C is the ?rst integral of equation (3.8a), refer

to (5.1));

(4.2)

[Q, H] = i?2 I, [Q, I] = [H, I] = 0

if the potential satis?es equation (3.8b);

[Q, H] = ?i?3 H (4.3)

if the potential satis?es equation (3.8c), and

[H, Q± ] = ±?Q± , (4.4a)

1

[Q+ , Q? ] = ? H 2 + (2? 2 + ?5 ) (4.4b)

48

if the potential satis?es (3.8d).

It follows from (4.1)–(4.3) that non-Lie SOs Q and Hamiltonians H form consistent

Lie algebras which can have rather nontrivial applications.

Formula (4.1b) presents an example of the general theorem [23, 24] stating that

commuting ordinary di?erential operators are connected by a polynomial algebraic

relation with constant coe?cients. In Section 7 we use relations (4.1) to integrate the

related equations (2.1).

Relations (4.2) de?ne the Heisenberg algebra. The linear combinations a± =

v (H ± iQ) realize the unusual representation of creation and annihilation operators

1

2

in terms of third-order di?erential operators.

In accordance with (4.3), Q plays a role of dilatation operator which continuously

changes eigenvalues of H. Indeed, let

(4.5)

H?E = E?E ,

then the function ? = exp(i?Q)?E (where ? is a real parameter) is also an eigen-

vector of the Hamiltonian H with the eigenvalue ?E.

It follows from (4.4) that for ?4 < 0 the operators Q+ and Q? are raising and

lowering operators for the corresponding Hamiltonian. In other words, if ?E sati-

s?es (4.5) then Q± ?E are also eigenfunctions of the Hamiltonian which, however,

correspond to the eigenvalues E ± ?:

H(Q± ?E ) = (E ± ?)(Q± ?E ). (4.6)

148 W.I. Fushchych, A.G. Nikitin

Relations (4.6) are typical for creation and annihilation operators of the quantum

oscillator. This observation shows a way for constructing exact solutions of the Schr?-

o

dinger equation whose potential satis?es relation (3.8d). Moreover, relations (4.4a)

allow Q to be interpreted as a conditional symmetry [8, 12]; such symmetries are of

particular interest in the analysis of partial di?erential equations [14, 25, 26]. Thus,

third-order SOs of equation (2.1) generate algebras of certain interest. Moreover,

algebraic properties of these SOs are the same for wide classes of potentials described

by one of equations (3.8).

5 Reduction of equations for potentials

Let us consider equations (3.8) in detail and describe the corresponding classes of

potentials. A solution of some of these nonlinear equations is a complicated problem

which, however, can be simpli?ed by using reductions to other well–studied equations.

5.a. The Weierstrass equation. Formula (3.8a) de?nes the Weierstrass equation

whose solutions are expressed via either elementary functions or via the Weierstrass

function, depending on values of the parameter ?1 and the integration constant.

Here we represent these well-known solutions (refer, e.g. to the classic monograph

of E.T. Whittaker and G.N. Watson [28]) in the form convenient for our purposes.

Multiplying the l.h.s. of (3.8a) by U and integrating we obtain

1

(U )2 ? U 3 + 3?1 U = C, (5.1)

2

where C is an integration constant which appeared above in (4.1b). Then by changing

roles of dependent and independent variables it becomes possible to integrate (5.1) and

to ?nd U as an implicit function of x. We will distinguish ?ve qualitatively di?erent

cases:

C 2 ? 4?1 = 0,

3

(5.2a)

C > 0,

C 2 ? 4?1 = 0,

3

(5.2b)

C < 0,

(5.2c)

C = ?1 = 0,

C 2 ? 4?1 < 0.

3

(5.3a)

C 2 ? 4?1 > 0.

3

(5.3b)

For (5.2a)–(5.2c), solutions of (5.1) can be expressed via elementary functions,

while (5.3a,b) generate solutions in elliptic functions.

For our purposes, it is convenient to transform (5.1) to another equivalent form.

Using the substitution

µ

U =V ? (5.4)

,

2

where µ is a real root of the cubic equation

µ3 ? 3?1 µ + C = 0, (5.5)

Higher symmetries and exact solutions of Schr?dinger equation

o 149

we obtain

1

(V )2 ? V 3 ? ?0 V 2 + 4? 1 V + 8? 0 ?1 = 0, (5.6)

? ? ??

2

where ?0 = 3 µ and ?1 = 3 (?1 ? µ2 ) are arbitrary real numbers.

? ?

2 4

The substitution (5.4), (5.5) transforms conditions (5.2), (5.3) to the following

form:

2

?1 ?1 ? ? 0

?2 (5.7a)

?? = 0, ?0 < 0,

?

2

?1 ? 1 ? ? 0

?2 (5.7b)

?? = 0, ?0 > 0,

?

2

?1 ? 1 ? ? 0

?2 (5.7c)

?? = 0, ?0 = 0,

?

?1 ?1 ? ?0 = 0,

?2 (5.8a)

?? ?1 > 0,

?

?1 ?1 ? ?0 = 0,

?2 (5.8b)

?? ?1 < 0.

?

?2

If relations (5.7a) are satis?ed, then ?1 = ?0 or ?1 = 0. Moreover, the correspon-

? ?

ding solutions for V di?er by a constant shift: V > V + 2? 0 , ?0 > ?0 /2. Without

?? ?

loss of generality we restrict ourselves to the former case, then solutions of equation

(5.6) corresponding to conditions (5.7a-c) have the following forms:

1 14

V = ? 2 2 tanh2 (?(x ? k)) ? 1 , ?0 = ? ? 2 , (5.9a)

? ?1 =

? ?,

2 4

1 14

V = ? 2 2 coth2 (?(x ? k)) ? 1 , ?0 = ? ? 2 , (5.9a )

? ?1 =

? ?,

2 4

12 14

V = ? 2 2 tan2 (?(x ? k) ? 1) , (5.9b)

?0 =

? ?, ?1 =

? ?,

2 4

2

(5.9c)

V= .

(x ? k)2

Here, k and ? are arbitrary real numbers.

For the cases (5.8) the general solution of (5.1) has the form

1

V = 2?(x ? k) + µ, (5.10)

2

where ? is a two-periodic Weierstrass function, which is meromorphic on all the

complex plane. The invariants of this function are g2 = ? 4 ?0 + 3? 1 and g3 =

2

3? ?

? 27 ?0 ?0 ? 9? 1 . Moreover, if condition (5.8a) holds, the corresponding solutions

4 2

?? ?

are bounded and can be expressed via the elliptic Jacobi functions

V = Bcn2 (Dx + k) + F, (5.11a)

where

B = (e3 ? e2 ), (e1 ? e3 )/2, (5.11b)

D= F = e2 ,

e1 > e2 > e3 are real solutions of the cubic equation from the r.h.s. of (5.6).

150 W.I. Fushchych, A.G. Nikitin

We note that formulae (5.9) present the set of well-known potentials which cor-

respond to the exactly solvable Schr?dinger equations [27]. In accordance with the

o

above, these equations admit extended Lie symmetries.

5.b. Painlev? and Riccati equations. Relation (3.8b) de?nes the ?rst Painlev?

e e

transcendent. Its solutions are meromorphic on all the complex plane but cannot be

expressed via elementary or special functions.

Equation (3.8c) is more complicated. However, by using the special change of

variables and applying the Miura [29] ansatz, we shall reduce it to the Painlev? form

e

also. Indeed, making the following change of variables

2

?3 1

3

U =? x=?3 (5.12)

V, y,

6 6?3

we obtain

1 2

+ V V ? xV ? V = 0, (5.13)

V V = ?V /?y.

3 3

The ansatz

1

V = W ? W2 (5.14)

6

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