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On the realization of the Weyl commutation relation HR = qRH
Physics Letters A 176 (1993) 300—302 North-Holland
PHYSICS LETTERS A
On the realization of the Weyl commutation relation HR = qRH A.S. Zhedanov I’hvsics Department, Doneisk University, Donetsk 340055, Ukraine Received 20 January 1993; accepted for publication 25 March 1993 (ommunicated b~A.P. Fordy
A generic solution of the Weyl relation !1R=qRH is found. whcrc II and Rare discrete tridiagonal Schrodinger operators ~iih real coefficients. This solution describes the two-parameter non-linear unitary aulonsorphism of the quantum SLq) 2) algebra This leads to a new class of exactly solvable discrete SchrOdinger equations having SL~2) as dsnamical synimctr~
The Weyl commutation relation HR=qRII
)
arises in the theory of the Heisenberg—Weyl group [1.21 as a special case of the Weyl identity with ii=exp(aa+fla~)
.
R=exp(;a+5a
)
a discrete basis. The operator I/ri ( 4a ) is chosen to he Hermitian in this basis. We propose also that 0< q < I . (~) We present a generic solution of (I ) given realiza-
tion (4) and show that the operators II. R and = R ~ form a non-trivial realization of the quailturn SL 5(2) algebra. Note that the operator!! in the
L q=exp(a5—
(2)
~‘).
where a. a ± are ordinary Bose operators and r~ . arc arbitrary constants. The operators (2) play also an important role in the theory of coherent states and theta functions [2]. On the other hand, relation (I ) arises also in the theory of quantum deformed algebras and groups
forrn (4a) can be considered as the discrete Schrödinger operator. So we obtain a new class of exactlr solvable discrete Schrodinger operators having SL,7(2) as a dynamical symmetry (cf. also ref. [5] ). Substituting (4) into (I) and equating the codt1cients in the terms ~. ii,, and u,, we obtain:
v’,,
[3]. The Weyl relation (1) plays a crucial role in Manin’s construction of “quantum planes” [4]. etc.
(i
) For
~i,,
2
ci,, = qa,, ci,,.+
An attractive feature of the commutation relations I ) is that the spectrum ).,, of the (Hermitian ) operator H has a “non-classical” behavior. ~,,
-~q”.
(3)
What are possible realizations of the Weyl relation (I) for the class of Hermitian operators H’? In this Letter we restrict ourselves to the problem of finding the operators II and R belonging to the class of tndiagonal operators,
fuji,,
=
R yt,, =
a,,~ d,, *
where a, 300
(4a I
+h,,i,ii,, + a,, iji,,_
~u,,± + g,, cu,, + r,,
vi,,
i
.
(4b)
r,, are some real coefficients and tjí,, is
a,,
,,,
=
qa,,
(6a
.
(6h
,,,
In what follows we assume that realization (4) is nondegenerate. i.e. a,,~0.d,,r,,~0.Then we obtain from (6) the following relations. ci,, =~, q “a,, . ~
=c2q~a,,.
(
7a 7h I
where c1. 2 are arbitrary real constants. (Of course. i,2 may be arbitrary functions in a with unit period c1 1(0 + I ) =12 (ii). However this dependence is “not observable” gisen realization (4).) (ii) For v,,, Elsevier Science Publishers 11.5
Volume 176, number 5
PHYSICS LETTERS A
(8a)
17 May 1993
c~=c
(17a) (17b)
1c2q~,
2(l+q)_i, (8b) Subtracting (8b) from (8a) we obtain the “integral” ~
(1 + q)g~
—
~~=_c3~4q i~=
(c~+c
5q’ ) (1
(Ci q —n + c
(9)
2 qfl~)bn = c3,
where c3 is a new arbitrary constant. From (8) and (9) one can easily find the solution for bn,
(1 7c)
2. +q)
Relations (16) define the quantum
SLq( 2)
algebra
[3,6,7]. Indeed, all the real forms of SLq(2) are de-
scribed by the commutation relations [81 [A0,A+]=±A..
(10)
n±2
(11) and C4 is a new arbitrary constant. The expression for g~is then obtained from (9). (iii) For cii,, we obtain the difference equation for w~=c1q~~C2q
a,,,
~
(18)
[A,A+]=uq~0+sqA0,
where
—v,,_1a~
—
(iii) “exp”-oscillator “cosh”-oscillatorfor forus=0 u=s>.0 [9]; (iv) [10,11].
q— 1
The Casimir operator for the SLq(2) algebra is
2,,+C
= —~-j-[(C1q
where u, s are arbitrary real parameters. The operator A0 is assumed to be Hermitian, whereas the operators A~ and A are Hermitian conjugates. The special cases of SL~(2)are: (i) SUq(2) for u= —s<0; (ii) SUq(l, I) for u=—s>’O;
+C2q”~)b
3b,,]
,
(12)
(19)
40)/(1_q).
~=A~A_
where
+(sq~~0_uq
Given the unitary representation of SLq(2) the n+ i
v,,=c 1q~—C2q
(13)
.
The general solution of (12) is 2v,,v,,_ i (14) W,~W,~_2 +C5 a,,2 = b,,b,,.1 (1 +q) where C 5 is a new arbitrary constant. So we have constructed a generic non-degenerate solution thecoefficients Weyl relation (4). The of five a~,b,,,(1) d,,, given g,,, r,, realization depend on five real parameters c~,..., C ‘
5.
In order to obtain the relation ofthis solution with known quantum algebras, let us introduce the operator L=R~,
Casimir takes the value Q= (sqk~uq~~o)/(1 —q)
(20)
,
resentation series. where the parameter k is to distinguish different repLet us fix the representation of SL~ (2) with the Casimir value
Q and introduce the operators 4°12A±
H= qA0,
,
L= A
— q’1°12
.
(21)
R = q’
By construction, the operator H is Hermitian and L =R
Using (19) one can obtain the relations 2 + QH+ uq/ (1 q), (22a) [s/ (1 q) ] H —
RL = — — LR=—[sq2/(l—q)]H2+qQH+uq/(1—q)
L~,,=r,,~ 1~~÷1+g,,y,,+d,,çv,,...1.
(15)
Using the explicit formulas (8)—( 14) for the coefficients a,, r,, one can directly verify that on the basis w~the products RL and LR are reduced to the quadratic forms in H, 2+~jH+(, (16a) RL=i~!-I LR=i~q2H2+,~qH+(, (16b)
(22b)
Relations (22) coincide with (16). So we can identify the operators (4), (1 5) with the SLq (2) operators (21). There is a correspondence between the arbitrary constants c~ c5 and the parameters of SLq(2) c 1c2=—sq/(l—q), (23a) c3c4_q(l+q)Q, (23b)
where 301
Volume 176, numberS
c~+ c, q
= q( I + q
PHYSICS LETTERS A
) 20/ ( I
—-j)
23c
.
(Oven the representation of SL
5(2). formulas I 231 impose three constraints to the five parameters U i~. This means that there are two f’ree parameters — sas, ri and c3 — which are independent of ihc representation parameters 11.5. C) of SL,J( 2). Therefore, sse have obtained a two—parameter isospectral family of the operators 1/. R. I. belonging to the same representation series of SL5( 2 1. In fact, one of these isospectral flows is trivial. Indeed, from I .23a ) ss c can parametrize
I’ \las
‘11
1 14) describe a ness class ,0 cxactl~solvable “dis crete potentials’’ (i.e. q—analogues of Morse, Poeschl
Teller. etc.. potentials) basing SL,, 2 1 as the dynanical symmetry (sec also ncf. [5] ). It is interesting to note that in ref. [12]. a ness class of potentials for the orcli,iai’r Schrödingcr equation was obtained with an cxp-Iike spectrum (3 ~. These polentials also base SL,( 2 ) as a dynamical svnlnletrv. However, it is not clear how the approach of ref. 121 can be related wiih Ihat pi’OPosed here ftc author thanks Pnol~ssor\ al. (iranos skii (on nlterestnig discussions at the topics treated in tli
sq ‘/( I —q) . 24 where s is a real parameter. This pararnetnizalion leads to translational invariance: the shift s s+ ~ is equivalent to the shift 11 ‘it—v1 in all the coefficients a, i,. The remaining one-parameter isospectral famils of the operators ii. R. L corresponds to non-trivial unitary cr—transformations of the initial SL,,( 2 ) generators. These q-transformations can he eonsidei’ed as q—analogues of’ linear automorphisms of’ ordinar\ Lie algcbnis As far as we know, such unitary (nonU=
papet.
References
928. AM l’ei,’loni,,, . ,ciir’ral,,,’d ,tih,’reni Oatr’s applications Springer Berlin 1’)SU
131 S
generators. . . . There is another interpretation of this result which mar he useful for physical applications. The operaton II (4a) can be considered as a discrete .Schrbdinger operator. Then the obtained solutions 1 1))).
302
Majid
4] ~ ~
nt I Mod I’hss ~
~Iuun.
))uantuin
I
5] R. Floreanini and I
I
~
I
I~ ~
~
,inr!
iii)fl-COiiifl1Ui,it.’
S net. I’Ii~s lent. 13 27
(I )S~ I Si,s Math 4) 1)955 I
ScOt
tOo:
111,2
955
I I vi limbo I ii NI oh I hs ..
~
99)0
S
~r’up
geomeirs preprinit Montreal
—~
linear) automorphisms for SL,,( — ) ai’e new. , So we have shown that all possible solutions ol the Weyl relation ( I ) in tennis of’ tridiagonal operators with real coefficients are isospectral flow’s of SL,,( 2 1
I
I-I. We~! I ruppcrithc~r,r’und ~
\ppl \iath
I Ytil ( r1iIlos skit. S S. Zhr’dan,,s Ph5s. [cii It 2~))I I’1’)2 115.
‘in
(,
$95 ‘1)1991)) 21)7
and (1.11 ( ~rakhio ~i,.1’
]‘i]ll.Yaii..).l’hss. \21)I’i’ilH I 055 [ID] 1.0 Biedenh~tri,,.1 Ph~s.S 25)195’) 1573: S 4 Maciarlane.J l’hss 5 2 19119) -15$) II] ~ ~ kulish and F V I)aniaskinsks
.1 l’hvs S 5
L4l s ]12]\’. Spiridonos . ‘hr ~. U’s kit
II ‘192 I 195
I
PHYSICS LETTERS A
On the realization of the Weyl commutation relation HR = qRH A.S. Zhedanov I’hvsics Department, Doneisk University, Donetsk 340055, Ukraine Received 20 January 1993; accepted for publication 25 March 1993 (ommunicated b~A.P. Fordy
A generic solution of the Weyl relation !1R=qRH is found. whcrc II and Rare discrete tridiagonal Schrodinger operators ~iih real coefficients. This solution describes the two-parameter non-linear unitary aulonsorphism of the quantum SLq) 2) algebra This leads to a new class of exactly solvable discrete SchrOdinger equations having SL~2) as dsnamical synimctr~
The Weyl commutation relation HR=qRII
)
arises in the theory of the Heisenberg—Weyl group [1.21 as a special case of the Weyl identity with ii=exp(aa+fla~)
.
R=exp(;a+5a
)
a discrete basis. The operator I/ri ( 4a ) is chosen to he Hermitian in this basis. We propose also that 0< q < I . (~) We present a generic solution of (I ) given realiza-
tion (4) and show that the operators II. R and = R ~ form a non-trivial realization of the quailturn SL 5(2) algebra. Note that the operator!! in the
L q=exp(a5—
(2)
~‘).
where a. a ± are ordinary Bose operators and r~ . arc arbitrary constants. The operators (2) play also an important role in the theory of coherent states and theta functions [2]. On the other hand, relation (I ) arises also in the theory of quantum deformed algebras and groups
forrn (4a) can be considered as the discrete Schrödinger operator. So we obtain a new class of exactlr solvable discrete Schrodinger operators having SL,7(2) as a dynamical symmetry (cf. also ref. [5] ). Substituting (4) into (I) and equating the codt1cients in the terms ~. ii,, and u,, we obtain:
v’,,
[3]. The Weyl relation (1) plays a crucial role in Manin’s construction of “quantum planes” [4]. etc.
(i
) For
~i,,
2
ci,, = qa,, ci,,.+
An attractive feature of the commutation relations I ) is that the spectrum ).,, of the (Hermitian ) operator H has a “non-classical” behavior. ~,,
-~q”.
(3)
What are possible realizations of the Weyl relation (I) for the class of Hermitian operators H’? In this Letter we restrict ourselves to the problem of finding the operators II and R belonging to the class of tndiagonal operators,
fuji,,
=
R yt,, =
a,,~ d,, *
where a, 300
(4a I
+h,,i,ii,, + a,, iji,,_
~u,,± + g,, cu,, + r,,
vi,,
i
.
(4b)
r,, are some real coefficients and tjí,, is
a,,
,,,
=
qa,,
(6a
.
(6h
,,,
In what follows we assume that realization (4) is nondegenerate. i.e. a,,~0.d,,r,,~0.Then we obtain from (6) the following relations. ci,, =~, q “a,, . ~
=c2q~a,,.
(
7a 7h I
where c1. 2 are arbitrary real constants. (Of course. i,2 may be arbitrary functions in a with unit period c1 1(0 + I ) =12 (ii). However this dependence is “not observable” gisen realization (4).) (ii) For v,,, Elsevier Science Publishers 11.5
Volume 176, number 5
PHYSICS LETTERS A
(8a)
17 May 1993
c~=c
(17a) (17b)
1c2q~,
2(l+q)_i, (8b) Subtracting (8b) from (8a) we obtain the “integral” ~
(1 + q)g~
—
~~=_c3~4q i~=
(c~+c
5q’ ) (1
(Ci q —n + c
(9)
2 qfl~)bn = c3,
where c3 is a new arbitrary constant. From (8) and (9) one can easily find the solution for bn,
(1 7c)
2. +q)
Relations (16) define the quantum
SLq( 2)
algebra
[3,6,7]. Indeed, all the real forms of SLq(2) are de-
scribed by the commutation relations [81 [A0,A+]=±A..
(10)
n±2
(11) and C4 is a new arbitrary constant. The expression for g~is then obtained from (9). (iii) For cii,, we obtain the difference equation for w~=c1q~~C2q
a,,,
~
(18)
[A,A+]=uq~0+sqA0,
where
—v,,_1a~
—
(iii) “exp”-oscillator “cosh”-oscillatorfor forus=0 u=s>.0 [9]; (iv) [10,11].
q— 1
The Casimir operator for the SLq(2) algebra is
2,,+C
= —~-j-[(C1q
where u, s are arbitrary real parameters. The operator A0 is assumed to be Hermitian, whereas the operators A~ and A are Hermitian conjugates. The special cases of SL~(2)are: (i) SUq(2) for u= —s<0; (ii) SUq(l, I) for u=—s>’O;
+C2q”~)b
3b,,]
,
(12)
(19)
40)/(1_q).
~=A~A_
where
+(sq~~0_uq
Given the unitary representation of SLq(2) the n+ i
v,,=c 1q~—C2q
(13)
.
The general solution of (12) is 2v,,v,,_ i (14) W,~W,~_2 +C5 a,,2 = b,,b,,.1 (1 +q) where C 5 is a new arbitrary constant. So we have constructed a generic non-degenerate solution thecoefficients Weyl relation (4). The of five a~,b,,,(1) d,,, given g,,, r,, realization depend on five real parameters c~,..., C ‘
5.
In order to obtain the relation ofthis solution with known quantum algebras, let us introduce the operator L=R~,
Casimir takes the value Q= (sqk~uq~~o)/(1 —q)
(20)
,
resentation series. where the parameter k is to distinguish different repLet us fix the representation of SL~ (2) with the Casimir value
Q and introduce the operators 4°12A±
H= qA0,
,
L= A
— q’1°12
.
(21)
R = q’
By construction, the operator H is Hermitian and L =R
Using (19) one can obtain the relations 2 + QH+ uq/ (1 q), (22a) [s/ (1 q) ] H —
RL = — — LR=—[sq2/(l—q)]H2+qQH+uq/(1—q)
L~,,=r,,~ 1~~÷1+g,,y,,+d,,çv,,...1.
(15)
Using the explicit formulas (8)—( 14) for the coefficients a,, r,, one can directly verify that on the basis w~the products RL and LR are reduced to the quadratic forms in H, 2+~jH+(, (16a) RL=i~!-I LR=i~q2H2+,~qH+(, (16b)
(22b)
Relations (22) coincide with (16). So we can identify the operators (4), (1 5) with the SLq (2) operators (21). There is a correspondence between the arbitrary constants c~ c5 and the parameters of SLq(2) c 1c2=—sq/(l—q), (23a) c3c4_q(l+q)Q, (23b)
where 301
Volume 176, numberS
c~+ c, q
= q( I + q
PHYSICS LETTERS A
) 20/ ( I
—-j)
23c
.
(Oven the representation of SL
5(2). formulas I 231 impose three constraints to the five parameters U i~. This means that there are two f’ree parameters — sas, ri and c3 — which are independent of ihc representation parameters 11.5. C) of SL,J( 2). Therefore, sse have obtained a two—parameter isospectral family of the operators 1/. R. I. belonging to the same representation series of SL5( 2 1. In fact, one of these isospectral flows is trivial. Indeed, from I .23a ) ss c can parametrize
I’ \las
‘11
1 14) describe a ness class ,0 cxactl~solvable “dis crete potentials’’ (i.e. q—analogues of Morse, Poeschl
Teller. etc.. potentials) basing SL,, 2 1 as the dynanical symmetry (sec also ncf. [5] ). It is interesting to note that in ref. [12]. a ness class of potentials for the orcli,iai’r Schrödingcr equation was obtained with an cxp-Iike spectrum (3 ~. These polentials also base SL,( 2 ) as a dynamical svnlnletrv. However, it is not clear how the approach of ref. 121 can be related wiih Ihat pi’OPosed here ftc author thanks Pnol~ssor\ al. (iranos skii (on nlterestnig discussions at the topics treated in tli
sq ‘/( I —q) . 24 where s is a real parameter. This pararnetnizalion leads to translational invariance: the shift s s+ ~ is equivalent to the shift 11 ‘it—v1 in all the coefficients a, i,. The remaining one-parameter isospectral famils of the operators ii. R. L corresponds to non-trivial unitary cr—transformations of the initial SL,,( 2 ) generators. These q-transformations can he eonsidei’ed as q—analogues of’ linear automorphisms of’ ordinar\ Lie algcbnis As far as we know, such unitary (nonU=
papet.
References
928. AM l’ei,’loni,,, . ,ciir’ral,,,’d ,tih,’reni Oatr’s applications Springer Berlin 1’)SU
131 S
generators. . . . There is another interpretation of this result which mar he useful for physical applications. The operaton II (4a) can be considered as a discrete .Schrbdinger operator. Then the obtained solutions 1 1))).
302
Majid
4] ~ ~
nt I Mod I’hss ~
~Iuun.
))uantuin
I
5] R. Floreanini and I
I
~
I
I~ ~
~
,inr!
iii)fl-COiiifl1Ui,it.’
S net. I’Ii~s lent. 13 27
(I )S~ I Si,s Math 4) 1)955 I
ScOt
tOo:
111,2
955
I I vi limbo I ii NI oh I hs ..
~
99)0
S
~r’up
geomeirs preprinit Montreal
—~
linear) automorphisms for SL,,( — ) ai’e new. , So we have shown that all possible solutions ol the Weyl relation ( I ) in tennis of’ tridiagonal operators with real coefficients are isospectral flow’s of SL,,( 2 1
I
I-I. We~! I ruppcrithc~r,r’und ~
\ppl \iath
I Ytil ( r1iIlos skit. S S. Zhr’dan,,s Ph5s. [cii It 2~))I I’1’)2 115.
‘in
(,
$95 ‘1)1991)) 21)7
and (1.11 ( ~rakhio ~i,.1’
]‘i]ll.Yaii..).l’hss. \21)I’i’ilH I 055 [ID] 1.0 Biedenh~tri,,.1 Ph~s.S 25)195’) 1573: S 4 Maciarlane.J l’hss 5 2 19119) -15$) II] ~ ~ kulish and F V I)aniaskinsks
.1 l’hvs S 5
L4l s ]12]\’. Spiridonos . ‘hr ~. U’s kit
II ‘192 I 195
I