Fac o iza ion o he hype geome ic- ype di e ence equa ion
on he non-uni o m la ices: dynamical algeb a
R. ´
Al a ez-Noda se†‡, N. M. A akishiye §and R. S. Cos as-San os∗
†Depa amen o de An´alisis Ma em´a ico.
Uni e sidad de Se illa. Apdo. 1160, E-41080 Se illa, Spain
‡Ins i u o Ca los I de F´ısica Te´o ica y Compu acional,
Uni e sidad de G anada, E-18071 G anada, Spain
§Ins i u o de Ma em´a icas, UNAM, Apa ado Pos al 273-3,
C.P. 62210 Cue na aca, Mo elos, M´exico
∗Depa amen o de Ma em´a icas, E.P.S., Uni e sidad Ca los III de Mad id.
A e. Uni e sidad 30, E-28911, Legan´es, Mad id, Spain
14 h Oc obe 2004
Abs ac
We a gue ha one can ac o ize he di e ence equa ion o hype geome ic ype on he non-
uni o m la ices in gene al case. I is shown ha in he mos cases o q-linea spec um
o he eigen alues his di ec ly leads o he dynamical symme y algeb a suq(1,1), whose
gene a o s a e explici ly cons uc ed in e ms o he di e ence ope a o s, ob ained in he
p ocess o ac o iza ion. Thus all models wi h he q-linea spec um (some o hem, bu no
all, p e iously conside ed in a numbe o publica ions) can be ea ed in a uni ied o m.
1 In oduc ion and p elimina ies
In his pape we con inue he s udy, s a ed in [1], on he ac o iza ion o he hype geome ic-
ype di e ence equa ion on he non-uni o m la ices, i.e., o he equa ion [2]
σ(s)∆
∆x(s−1
2)∇y(s)
∇x(s)+τ(s)∆y(s)
∆x(s)+λy(s) = 0,
σ(s) = ˜σ(x(s)) −1
2˜τ(x(s))∆xs−1
2, τ(s) = ˜τ(x(s)),
(1)
whe e ∆y(s) := y(s+ 1) −y(s), ∇y(s) := y(s)−y(s−1), ˜σ(x(s)) and ˜τ(x(s)) a e polynomials
in x(s) o deg ee a mos 2 and 1, espec i ely, and λis a cons an (see also [3]). The di e ence
equa ion (1) has polynomial solu ions Pn(x(s); q):= Pn(s;q) o he hype geome ic ype i and
only i he la ice x(s) has he o m [4, 5]
x(s) = c1(q)qs+c2(q)q−s+c3(q) = c1(q)[qs+q−s−µ] + c3(q),(2)
whe e c1,c2,c3and qµ:= c1/c2a e cons an s which, in gene al, depend on q. An impo an
special case o he la ice x(s) is he q-linea la ice, which is ob ained om (2) by assuming
ha ei he c1(q) o c2(q) anishes.
1
The polynomial solu ions o he di e ence equa ion (1) co espond o he ollowing exp ession
[3] o i s eigen alues λn(q):
λn(q) = C1qn+C2q−n+C3,
C1=1
2(1 −q)eτ0+eσ00
kq, C2=1
2(1 −q−1)eτ0−eσ00
kq, C3=−eσ00(1 + q)
2kq(1 −q)−eτ0
2,
(3)
whe e eτ0and eσ00 a e he coe icien s o x(s) and x2(s) in he Taylo expansion o ˜τ(x(s)) and
˜σ(x(s)), espec i ely, i.e., ˜τ(x(s)) = eτ0x(s) + eτ(0), and ˜σ(x(s)) = eσ00/2x2(s) + eσ0(0)x(s) + eσ(0).
Obse e ha he coe icien s C1and C2o he qnand q−n e ms, espec i ely, a e ixed by
he unc ions σand τin (1), and so is he p oduc C1C2. In wha ollows we deno e by Lq he
alue o C1C2=(eσ00/kq)2−(eτ0)2/4k2
q.
The sequence {λn(q)}sa is ies he ollowing h ee- e m ecu ence ela ion (TTRR)
λn+2(q)−(q+q−1)λn+1(q) + λn(q) = 1
2(˜τ0k2
q−˜σ00[2]q) = C. (4)
Con e sely, i {λn(q)}sa is y he TTRR (4), hen i has he o m λn(q) = C0
1qn+C0
2q−n+C0
3.
Ob iously, ha ing used he ini ial condi ions λ0(q) = 0 and λ1(q) = −eτ0, one eco e s he ex-
p ession (3).
I is well known [3] ha unde ce ain condi ions he polynomial solu ions o (1) a e o hog-
onal. Fo example, i σ(s)ρ(s)xk(s−1
2)s=a,b = 0, o all k= 0,1,2,..., hen he polynomial
solu ions possess a disc e e o hogonali y p ope y
b−1
X
s=a
Pn(s;q)Pm(s;q)ρ(s)∇x1(s) = d2
n(q)δn,m,(5)
whe e he weigh unc ion ρ(s) is a solu ion o he Pea son- ype di e ence equa ion [3]
∆
∆x(s−1
2)[σ(s)ρ(s)] = τ(s)ρ(s) o σ(s+ 1)ρ(s+ 1) = σ(−s−µ)ρ(s).(6)
I he la ice x(s) is a q-linea la ice, i.e., x(s) = c q±s+c3, hen he σ(−s−µ) in (6) should be
subs i u ed by σ(s) + τ(s)∆x(s−1/2). A mo e de ailed in o ma ion on o hogonal polynomials
on he non-uni o m la ices can be ound in [3, 5, 6, 7, 8].
In [1] i has been shown ha one can ac o ize he Niki o o -U a o equa ion (1) wi h he
aid o aising and lowe ing ope a o s, which can be cons uc ed o solu ions o his equa ion.
In his pape we wish o make one s ep u he by s udying he dynamical symme y alge-
b a o he hype geome ic- ype di e ence equa ion (1) on he non-uni o m la ices (2). Ou
app oach is essen ially based on he simple obse a ion, o mula ed in [9]: In o de o ac o -
ize an a bi a y di e ence equa ion, one should exp ess i explici ly in e ms o he shi (o
displacemen ) ope a o s exp(ad
ds), which a e de ined as exp(ad
ds) (s) = (s+a), ais some
cons an . Fo example, in he case o he equa ion (1) his co esponds o he subs i u ions
∆ = exp( d
ds )−1 and ∇= 1 −exp(−d
ds). This p ocedu e con e s a di e ence equa ion in o an
eigen alue p oblem o a di e ence ope a o , ep esen ed by a linea combina ion o some shi
ope a o s (wi h coe icien s, which depend polynomially on he a iable s). Since each e m o
his linea combina ion is eadily ac o izable (because exp (α+β)A= exp α A exp β A o an
a bi a y ope a o A), he ac o iza ion o he whole linea combina ion, which ep esen s he
2
ini ial di e ence equa ion, becomes s aigh o wa d.
Inspi ed by he appea ance o Mac a lane’s [10] and Biedenha n’s [11] impo an cons uc-
ions o q-analogues o quan um ha monic oscilla o , his echnique o ac o iza ion o di e ence
equa ions was la e employed in a numbe o publica ions [12]–[16] in o de o s udy g oup
heo e ic p ope ies o he a ious well-known amilies o o hogonal polynomials, which can be
iewed as q-ex ensions o he classical He mi e polynomials. So ou pu pose he e is o o mu-
la e a uni ied app oach o de i ing all o hese esul s, which co espond o he q-linea spec um.
An impo an aspec o obse e a his poin is ha we shall mainly (excep o he exam-
ples in subsec ion 4.2) con ine ou a en ion o hose amilies o q-polynomials, which sa is y
disc e e o hogonali y ela ion o he ype (5). The explana ion o such p e e ence is ha he
ac o iza ion o di e ence equa ions o ins ances o q-polynomials wi h con inuous o hogonali y
p ope y has been al eady ho oughly s udied in [12]–[16]. Obse e also ha ou app oach s ill
emains alid in he limi as q→1; so classical coun e pa s o q-polynomials, which will be
discussed in his pape , a e in ac inco po a ed as app op ia e limi cases. Bu he eade who
desi es o know mo e abou he ac o iza ion in he cases o classical o hogonal polynomials
(such as he K a chuk, Cha lie , Meixne , Meixne –Pollaczek, and Hahn) may be e e ed o
[17, 18] and e e ences he ein.
The pape is o ganized as ollows. In sec ion 2 we associa e wi h each amily o q-polynomials
a “q-Hamil onian” H(s;q) ( ia he second-o de di e ence equa ion) and cons uc wo di e ence
ope a o s a(s;q) and b(s;q), which ac o ize he ope a o H(s;q). Ou main esul s a e gi en in
sec ion 3: hey a e o mula ed in Theo ems 3.4 – which gi es a simple necessa y and su icien
condi ion ha he q-Hamil onian H(s;q) admi s he ac o iza ion in e ms o he ope a o s
a(s;q) and b(s;q), which sa is y he ela ion a(s;q)b(s;q)−qγb(s;q)a(s;q) = I o some γ, and
3.5 – s a ing ha he eigen alues o he di e ence equa ion (1) in his case should be o he o m
λn(q) = C1qn+C3o λn(q) = C2q−n+C3. In sec ion 4 se e al ele an examples o pa icula
q- amilies o o hogonal polynomials a e illus a ed.
2 Fac o iza ion ope a o s
Le in oduce a se o unc ions Φn
Φn(s;q) = d−1
nA(s)pρ(s)Pn(s;q),(7)
whe e dnis he no m o he q-polynomials Pn(s;q), ρ(s) is he solu ion o he Pea son equa ion
(6) and A(s) is an a bi a y con inuous unc ion, A(s)6= 0 in he in e al (a, b) o o hogonali y
o Pn. I he polynomials Pn(s;q) possess he disc e e o hogonali y p ope y (5), hen he
unc ions Φn(s;q) sa is y
hΦn(s;q),Φm(s;q)i=
b−1
X
s=a
Φn(s;q)Φm(s;q)∇x1(s)
A2(s)=δn,m.(8)
No ice ha i A(s) = p∇x1(s), hen he se (Φn)nis an o hono mal se . Ob iously, in he
case o a con inuous o hogonali y (as o he Askey-Wilson polynomials) one needs o change
he sum in (8) by a Riemann in eg al [3, 5].
Nex , we de ine he q-Hamil onian H(s;q) o he o m
H(s;q) := 1
∇x1(s)A(s)H(s;q)1
A(s),(9)
3
whe e
H(s;q) := −pσ(−s−µ+1)σ(s)
∇x(s)e−∂s−pσ(−s−µ)σ(s+ 1)
∆x(s)e∂s+σ(−s−µ)
∆x(s)+σ(s)
∇x(s)I, (10)
eα∂s (s) = (s+α) o all α∈Cand Iis he iden i y ope a o . I we now use he iden i y
∇= ∆ −∇∆ and he equa ion (1), we ind ha
H(s;q)Φn(s;q) = λnΦn(s;q),(11)
i.e., he unc ions Φn(s;q), de ined in (7), a e he eigen unc ions o he associa ed ope a o
H(s;q).
Ou i s s ep is o ind wo ope a o s a(s;q) and b(s;q) such ha he Hamil onian H(s;q) =
b(s;q)a(s;q), i.e., he ope a o s a(s;q) and b(s;q) ac o ize he q-Hamil onian H(s;q). Bu
be o e exhibi ing hei explici o m le us poin ou ha i he e exis s a pai o such ope a o s,
hen he e a e in ini ely many o hem. Indeed, le a(s;q) and b(s;q) be such ope a o s ha
H(s;q) = b(s;q)a(s;q) and le U(s;q) be an a bi a y uni a y ope a o , i.e., U†(s;q)U(s;q) = I.
Then he ope a o s
ea(s;q) := U(s;q)a(s;q),eb(s;q) := b(s;q)U†(s;q),
also ac o ize H(s;q) o
eb(s;q)ea(s;q) = b(s;q)U†(s;q)U(s;q)a(s;q) = b(s;q)I a(s;q) = b(s;q)a(s;q) = H(s;q).
This a bi a iness in picking up a pa icula uni a y ope a o U(s) is e y essen ial because
i enables one o cons uc a closed algeb a, which con ains a Hamil onian H(s;q) i sel . An
explici o m o he spec um o his Hamil onian may hen be ound by pu ely algeb aic a gu-
men s om he knowledge o ep esen a ions o his algeb a (which is he e o e e e ed o as a
dynamical algeb a).
I one applies he s anda d ac o iza ion p ocedu e o he equa ion (1), hen he ollowing di -
e ence ope a o s eme ge
De ini ion 2.1 Le αbe a eal numbe and A(s)an a bi a y con inuous non- anishing unc ion
in (a, b). We de ine a amily o α-down and α-up ope a o s by
a↓
α(s;q):= A(s)
p∇x1(s)e−α∂s e∂ssσ(s)
∇x(s)−sσ(−s−µ)
∆x(s)!1
A(s),
a↑
α(s;q):= 1
∇x1(s)A(s) sσ(s)
∇x(s)e−∂s−sσ(−s−µ)
∆x(s)!eα∂sp∇x1(s)
A(s),
(12)
espec i ely.
A s aigh o wa d calcula ion (by using he simple iden i y e∂s∇= ∆) shows ha o all α∈R
H(s;q) = a↑
α(s;q)a↓
α(s;q),(13)
i.e., he ope a o s a↓
α(s;q) and a↑
α(s;q) ac o ize he Hamil onian, de ined in (9). i.e., we ha e
he ollowing
Theo em 2.2 Gi en a q-Hamil onian (9) H(s;q), hen he ope a o s a↓
α(s;q)and a↑
α(s;q)de-
ined in (12) a e such ha o all α∈C,H(s;q) = a↑
α(s;q)a↓
α(s;q).
4
Ou nex s ep is o ind a dynamical algeb a, associa ed wi h he Hamil onian H(s;q). To his
end we will need he ollowing de ini ion
De ini ion 2.3 A unc ion (z)is said o be a linea - ype unc ion o z, i he e exis wo
unc ions Fand G, such ha o all z, ζ ∈C, his unc ion (z)can be ep esen ed as
(z+ζ) = F(ζ) (z) + G(ζ).
A pa icula case o he linea - ype unc ions a e he q-linea unc ions, i.e., he unc ions o he
o m (z) = Aqz+B. Fo hese unc ions (z+ζ) = F(ζ) (z) + G(ζ), whe e F(ζ) = qζand
G(ζ) = B(1 −qζ).
Rema k 2.4 I we use he exp ession in (3) o he eigen alues λn, hen i is s aigh o wa d
o see ha λnis a q-linea unc ion o ni and only i eσ00 =±kqeτ0. Mo eo e , in his case we
ha e
eσ00 =kqeτ0⇒λn(q, +) = eτ0
1−q(qn−1),and eσ00 =−kqeτ0⇒λn(q, −) = eτ0
1−q−1(q−n−1).
(14)
No ice ha λn(q, −) = λn(q−1,+), i.e., he second case can be ob ained o m he i s one jus
by changing q o q−1.
P oposi ion 2.5 The unc ion λnis a q-linea unc ion o ni and only i i sa is ies λn+1 =
qλn+C.
P oo : A s aigh o wa d compu a ions show ha i λnis a q-linea unc ion o n, hen i
sa is ies he ecu ence o mula λn+1 =qλn+C, whe e Cis a cons an (in his case C=λ1).
Bu he gene al solu ion o he di e ence equa ion λn+1 =qλn+Cis λn=Aqn+D, whe e A
and Da e, in gene al, non- anishing cons an s.
Rema k 2.6 No ice ha i λnis a q-linea unc ion o n, hen λnsa is ies he ecu ence
ela ion λn+γ−qγλn=C o any numbe s γand C.
Finally, we ha e he ollowing s aigh o wa d lemma
Lemma 2.7 Le x(s)be a q-linea unc ion o sand λnbe he eigen alue o he di e ence equa-
ion o hype geome ic ype (1). Then λnis a q-linea unc ion o ni and only i ∆(2)(σ(s)) = 0
and q−1-linea unc ion o ni and only i ∆(2)(σ(−s−µ)) = 0, whe e ∆(2) is he ope a o
∆(2) =∆
∆x1(s)
∆
∆x(s).
P oo : I ollows om equa ion (14) and he ac ha ∆(2)(σ(s)) = [2]q
2(eσ00 −˜τ0kq) and
∆(2)(σ(−s−µ)) = [2]q
2(eσ00 + ˜τ0kq).
3 Dynamical algeb a
We begin his sec ion wi h he ollowing de ini ion.
De ini ion 3.1 Le ςbe a eal numbe , and le a(s;q)and b(s;q)be wo ope a o s. We de ine
he ς-commu a o o aand bas
[a(s;q), b(s;q)]ς=a(s;q)b(s;q)−ςb(s;q)a(s;q).
5

P oposi ion 3.2 Le H(s;q)be an ope a o , such ha he e exis wo ope a o s a(s;q)and
b(s;q)and wo eal numbe s ςand Λ, such ha H(s;q) = b(s)a(s;q), and [a(s;q), b(s;q)]ς= Λ.
Then, i Φ(s;q)is an eigen ec o o he Hamil onian H(s;q), associa ed wi h he eigen alue λ,
we ha e
1. H(s;q){a(s;q)Φ(s;q)}=ς−1(λ−Λ) {a(s;q)Φ(s;q)}, i.e., a(s;q)Φ(s;q)is he eigen ec o
o H(s;q), associa ed wi h he eigen alue ς−1(λ−Λ),
2. H(s;q){b(s;q)Φ(s;q)}= (Λ + ςλ){b(s;q)Φ(s;q)}, i.e., b(s;q)Φ(s;q)is he eigen ec o o
H(s;q), associa ed wi h he eigen alue Λ + ςλ.
P oo : In he i s case, since H(s;q)Φ(s;q) = λΦ(s;q),
H(s;q){a(s;q)Φ(s;q)}=b(s;q)a(s;q){a(s;q)Φ(s;q)}=ς−1(a(s;q)b(s;q)−Λ){a(s;q)Φ(s;q)}
=ς−1(λ−Λ){a(s;q)Φ(s;q)}.
By he same oken, in he second case
H(s;q){b(s;q)Φ(s;q)}=b(s;q)a(s;q)b(s;q)Φ(s;q) = b(s;q)(Λ + ςλ)Φ(s;q)
= (Λ + ςλ){b(s;q)Φ(s;q)}.
In he same way one can p o e ha
a(s;q)b(s;q)Φ(s;q) = (Λ + ςλ)Φ(s;q).(15)
Mo eo e , i Φ(s;q) is an eigen ec o o he Hamil onian H(s;q) (o o he ope a o a(s;q)b(s;q)),
hen ak(s;q)Φ(s;q) and bk(s;q)Φ(s;q) a e, in gene al, also eigen ec o s.
Rema k 3.3 Ob iously, he condi ion [a(s;q), b(s;q)]ς=Ican be changed o [a(s;q), b(s;q)]ς=
Λ, whe e Λis an a bi a y non-ze o cons an . In ac , i he ope a o s a(s;q)and b(s;q)sa is y
he q-commu a ion ela ion [a(s;q), b(s;q)]ς= Λ, hen he ope a o s a(s;q) = Λ−1/2a(s;q)and
b(s;q) = Λ−1/2b(s;q)sa is y [a(s;q),b(s;q)]ς=I, and H(s;q) = Λb(s;q)a(s;q).
The P oposi ion 3.2 hus e e s o he case o a sys em, desc ibed by a Hamil onian H(s;q),
which admi s he ac o iza ion (13) in e ms o he ope a o s a(s;q) and b(s;q), sa is ying he
q-commu a ion ela ion [a(s;q), b(s;q)]ς=I. Mo eo e , i ells us how o cons uc a dynam-
ical symme y algeb a o such a case in a di ec ashion [19]. Indeed, le us assume ha
[a(s;q), b(s;q)]ς=I, ς =q2(o q−2), and b(s;q) = a†(s;q). Then one can ew i e he q2-
commu a o a(s;q)a†(s;q)−q2a†(s;q)a(s;q) = Iin he ollowing o m
[a(s;q), a†(s;q)] := a(s;q)a†(s;q)−a†(s;q)a(s;q) = I−(1 −q2)a†(s;q)a(s;q) := q2N(s),
whe e, by de ini ion, he ope a o N(s) is equal o N(s) = ln[I−(1 −q2)a†(s;q)a(s;q)]/ln q2.
F om his de ini ion o N(s) i ollows ha
[N(s), a(s;q)] = −a(s;q),[N(s), a†(s;q)] = a†(s;q),(16)
i.e., N(s) is he numbe ope a o . The nex (and inal) s ep is o in oduce a new se o he
ope a o s
b(s;q) := q−N(s)/2a(s;q), b†(s;q) := a†(s;q)q−N(s)/2,
which sa is y he ollowing commu a ion ela ion
b(s;q)b†(s;q)−q b†(s;q)b(s;q) = q−N(s),
6
eadily de i ed wi h he aid o (16). The ope a o s b(s;q), b†(s;q), and N(s) di ec ly lead o
he dynamical algeb a suq(1,1) wi h he gene a o s
K0(s) = 1
2[N(s) + 1/2], K+(s) = β(b†(s;q))2, K−(s) = β b2(s;q), β−1=q+q−1.
I is s aigh o wa d o e i y ha hus de ined gene a o s sa is y he s anda d commu a ion
ela ions
[K0(s), K±(s)] = ±K±(s),[K−(s), K+(s)] = [2K0(s)]q2,
o he algeb a suq(1,1) (see e.g. [20]).
Thus in he case when he ope a o s a(s;q) and b(s;q), which ac o ize he Hamil onian
H(s;q), sa is y he q-commu a ion ela ion [a(s;q), b(s;q)]ς2=Iand b(s;q) = a†(s;q), he ap-
p op ia e dynamical symme y algeb a is suς(1,1). So he ques ion a ises: wha a e condi ions
o insu ing ha such q-commu a o akes place? In o he wo ds, we ha e he ollowing
P oblem 1: To ind wo ope a o s a(s;q)and b(s;q)and a cons an ςsuch ha he Hamil onian
H(s;q) = b(s;q)a(s;q)and [a(s;q), b(s;q)]ς=I.
Fo he i s pa we al eady ha e he answe (see Theo em 2.2). The solu ion o he second
one is o mula ed in he ollowing wo heo ems.
Theo em 3.4 Le H(s;q)be he ollowing di e ence ope a o (q-Hamil onian)
H(s;q) = 1
∇x1(s)A(s)H(s;q)1
A(s).(17)
The ope a o s b(s;q) = a↑
α(s;q)and a(s;q) = a↓
α(s;q)gi en in (12) ac o ize he Hamil onian
H(s;q)(17) and sa is y he commu a ion ela ion [a(s;q), b(s;q)]ς= Λ o a ce ain eal numbe
ςi and only i he ollowing wo condi ions hold:
∇x(s)
∇x1(s−α)s∇x1(s−1)∇x1(s)
∇x(s−α)∆x(s−α)sσ(s−α)σ(−s−µ+α)
σ(s)σ(−s−µ+ 1) =ς, (18)
and
1
∆x(s−α)σ(s−α+ 1)
∇x1(s−α+ 1) +σ(−s−µ+α)
∇x1(s−α)−ς1
∇x1(s)σ(s)
∇x(s)+σ(−s−µ)
∆x(s)= Λ.(19)
P oo : Taking he exp ession o he ope a o s a↑
α(s) and a↓
α(s), a s aigh o wa d calculus shows
ha a↓
α(s)a↑
α(s) = A1(s)e∂s+A2(s)e−∂s+A3(s)I, whe e
A1(s) = −s∇x1(s+ 1)
∇x1(s)
A(s)
A(s+ 1)sσ(s+ 1 −α)σ(−s−µ−1 + α)
∆x(s−α)∆x(s+ 1 −α)
1
∇x1(s+ 1 −α),
A2(s) = −s∇x1(s−1)
∇x1(s)
A(s)
A(s−1)sσ(s−α)σ(−s−µ+α)
∆x(s−1−α)∆x(s−α)
1
∇x1(s−α),
A3(s) = 1
∆x(s−α)σ(s+ 1 −α)
∇x1(s+ 1 −α)+σ(−s−µ+α)
∇x1(s−α).
(20)
7
In he same way, using (10) and (9) we ha e a↑
α(s)a↓
α(s) = H(s;q) = B1(s)e∂s+B2(s)e−∂s+
B3(s)I, whe e
B1(s) = −1
∇x1(s)
A(s)
A(s+ 1) pσ(−s−µ)σ(s+ 1)
∇x(s+ 1) ,
B2(s) = −1
∇x1(s)
A(s)
A(s−1) pσ(−s−µ+ 1)σ(s)
∇x(s),
B3(s) = 1
∇x1(s)σ(s)
∇x(s)+σ(−s−µ)
∆x(s).
(21)
Consequen ly,
[a↓
α(s), a↑
α(s)]ς=A1(s)−ςB1(s)e∂s+A2(s)−ςB2(s)e−∂s+A3(s)−ςB3(s)I. (22)
To elimina e he wo e ms in he igh -hand side o (22), which a e p opo ional o he di e ence
ope a o s exp(±∂s), one mus equi e ha
A1(s)−ς B1(s) = 0, A2(s)−ς B2(s) = 0.(23)
O hand, i is no e iden ha one can sa is y bo h o he ela ions (23), which only in ol e he
same cons an ς. Bu i is s aigh o wa d o e i y om (20) and (21) ha
A1(s)B2(s+ 1) = A2(s+ 1) B1(s),
o , equi alen ly,
A1(s)
B1(s)=A2(s+ 1)
B2(s+ 1).
Hence, he equi emen ha A1(s) = ς B1(s) en ails he ela ion A2(s) = ς B2(s), and ice e sa.
F om (22) i is now e iden ha he commu a o [a↑
α(s), a↓
α(s)]ςis a cons an i (23) holds and
he ac o A3(s)−ς B3(s) is a cons an . Thus, he equi ed condi ions (18) and (19) immedia ely
ollow. 
Theo em 3.5 Le (Φn)n he eigen unc ions o H(s;q)co esponding o he eigen alues (λn)n
and suppose ha he p oblem 1 has a solu ion o Λ6= 0. Then, he eigen alues λno he
di e ence equa ion (11) a e q-linea o q−1-linea unc ions o n, i.e., λn=C1qn+C3o
λn=C2q−n+C3, espec i ely.
P oo : 1In he ollowing we use he no a ion ς=qγ. Suppose ha p oblem 1 has a solu ion wi h
Λ6= 0 and λnis no a q-linea ( espec i ely, q−1) unc ion o n. F om P oposi ion 3.2 we know
ha a↑
α(s;q)Φn(s;q) is and eigen ec o o H(s;q) co esponding o he eigen alue Λ+qγλn. I we
deno e by Φm(n);qsuch eigen ec o whe e m(n) is a unc ion o n, hen we ha e Λ+qγλn=λm(n).
Then using (3) we ge
λm(n)=C1qm(n)+C2q−m(n)+C3, C1C2=Lq.(24)
On he o he hand
λm(n)= Λ + qγλn=C1qγqn+C2qγq−n+qγC3+ Λ = C0
1qn+C0
2q−n+C0
3.(25)
Bu he e, since λm(n)is an eigen alue o (1), again we ha e he condi ion C0
1C0
2=Lq, hus
C1C2=Lq=C0
1C0
2=C1C2q2γso q2γ= 1, i.e., γ= 0, o C1C2= 0. In he i s case, equa ing
1Fo an al e na i e p oo in he case α= 0 see he appendix.
8
(24) and (25), we ha e ha C0
3=C3qγ+ Λ = C3, i.e., Λ = 0 ha is a con adic ion. Thus
C1C2= 0 om whe e he esul easily ollows.
I is wo h no ing ha he q-linea i y o he eigen alues o H(s;q) is only he necessa y condi ion,
i.e., i is no su icien . So he e a e cases when he eigen alues λn(q) a e q-linea ( o ins ance,
hose which co espond o he q-Meixne , he q-Cha lie , and he q-Lague e polynomials wi h
a6=q−1/2), bu he co esponding q-Hamil onians H(s;q) do no admi he ac o iza ion in e ms
o q-commu ing ope a o s. This jus e lec s he ac ha an app op ia e dynamical algeb a is
no suq(1,1) and one has o conside a mo e complica ed quad a ic algeb a AW(3) [21]. The
p oblem o inding an explici connec ion be ween he gene a o s o he algeb a AW(3) and he
ope a o s a(s;q) and b(s;q), which ac o ize he q-Hamil onians o hese cases, will be a ended
in a sepa a e publica ion.
Rema k 3.6 A special impo an case o he non-linea la ice is when x(s) = 1
2(qs+q−s). In
his case i we pu α=1
2, hen he condi ions (18) and (19) o Theo em 3.4 becomes
sσ(s−1
2)σ(−s+1
2)
σ(s)σ(−s+ 1) =ς,
and
1
∇x1(s) σ(s+1
2)
∆x(s)+σ(−s+1
2)
∇x(s)!−ς1
∇x1(s)σ(s)
∇x(s)+σ(−s)
∆x(s)= Λ.
espec i ely. Mo eo e , i we pu A(s) = 1 hen, H(s;q) = (∇x1(s))−1H(s;q)and he α-ope a o s
simpli y
a↓
1/2(s) = 1
∇x1(s)e1
2∂spσ(s)−e−1
2∂spσ(−s),
a↑
1/2(s) = 1
∇x1(s)pσ(s)e−1
2∂s−pσ(−s)e1
2∂s.
Now we can o mula e he
P oblem 2: To ind wo ope a o s a(s;q)and b(s;q)and a cons an ςsuch ha he Hamil onian
H(s;q) = b(s;q)a(s;q)and [a(s;q), b(s;q)]ς=Iand such ha a(s;q)and b(s;q)a e he lowe ing
and aising ope a o s, i.e.,
a(s;q)Φn(s;q) = DnΦn−1(s;q) and b(s;q)Φn(s;q) = UnΦn+1(s;q).(26)
Again, wi hou loss o gene ali y, we will change he condi ion [a(s;q), b(s;q)]ς=Iin o
[a(s;q), b(s;q)]ς= Λ and chose Λ = λ1.
Also he ope a o s b(s;q) = a↑
α(s;q) and a(s;q) = a↓
α(s;q), gi en in (12), p o ide he ac o iza ion
o H(s;q).
I we now apply b(s;q) o he i s equa ion o (26) and use he second one as well as (11), we
ind ha λn=DnUn−1. On he o he hand, applying a(s;q) o he second equa ion in (26) and
using he i s one as well as (15) we ob ain λ1+ςλn=UnDn+1 =λn+1. Thus, using P oposi ion
2.5 we conclude ha λnshould be a ς-linea unc ion, i.e., λnhas he o m λn=Aςn+D, whe e
Aand Da e non- anishing cons an s. Mo eo e , using he ecu ence λ1+ςλn=λn+1 and
P oposi ion 3.2 we ob ain ha
H(s;q){a(s;q)Φn(s;q)}=λn−1{a(s;q)Φn(s;q)},
H(s;q){b(s;q)Φn(s;q)}=λn+1{b(s;q)Φn(s;q)},
9
B(x) = c qx(1 −b qx+1), D(x) = (1 −qx)(1 + bc qx).
So q-Meixne unc ions, de ined as
ΦM
n(x;b, c;q) := d−1
n(b, c)cx(bq;q)x
(q;q)x(−bcq;q)x1/2
qx(x−1)/4Mn(q−x;b, c;q),
a e eigen unc ions o a di e ence “Hamil onian” HM(x;b, c;q),
HM(x;b, c;q)ΦM
n(x;b, c;q) = 1−qn
1−qΦM
n(x;b, c;q),
whe e
HM(x;b, c;q) := 1
1−qhB(x) + D(x)−B1/2(x)e∂xD1/2(x)−D1/2(x)e−∂xB1/2(x)i.
The q-Meixne unc ions sa is y he disc e e o hogonali y ela ion
∞
X
k=0
ΦM
m(k;b, c;q) ΦM
n(k;b, c;q) = δmn .
One can ac o ize he “Hamil onian” HM(x;b, c;q),
HM(x;b, c;q) = a↑
M(x;b, c;q)a↓
M(x;b, c;q),
by means o he “lowe ing” and “ aising” di e ence ope a o s
a↓
M(x;b, c;q) := 1
√1−qhe∂xD1/2(x)−B1/2(x)i,
a↑
M(x;b, c;q) := 1
√1−qhD1/2(x)e−∂x−B1/2(x)i.
(29)
The di e ence ope a o s (29) sa is y a q-commu a ion ela ion o he o m
a↓
M(x;b/q, cq;q)a↑
M(x;b/q, cq;q)−qa↑
M(x;b, c;q)a↓
M(x;b, c;q) = I .
Thei ac ion on he q-Meixne unc ions is gi en by
a↓
M(x;b, c;q) ΦM
n(x;b, c;q) = q1−qn
1−qΦM
n−1(x;bq, c/q;q),
a↑
M(x;b/q, cq;q) ΦM
n(x;b, c;q) = q1−qn+1
1−qΦM
n+1(x;b/q, cq;q),
(30)
ha is, hey no only lowe and aise, espec i ely, he index n, bu al e also he pa ame e s b
and c. The o mulae (30) a e equi alen o he s a emen ha he o wa d and backwa d shi
ope a o s o he q-Meixne polynomials (28) ha e he o m (see [7], p.95, (3.13.2) and (3.13.8))
(1 −e∂x)Mn(q−x;b, c;q) = 1−qn
c(1 −bq)q−xMn−1(q−x;bq, c/q;q),
hcqx(1 −bqx)−(1 −qx)(1 + bcqx)e−∂xiMn(q−x;b, c;q)
=cqx(1 −b)Mn+1(q−x;b/q, cq;q).
I emains only o emind he eade ha when he pa ame e bin (28) anishes, he q-
Meixne polynomials Mn(q−x; 0, c;q) coincide wi h he q-Cha lie polynomials ([7], p.112)
Cn(q−x;c;q) := 2φ1
q−n, q−x
q;−qn+1
c
0
.(31)
In his case B(x) = c qxand D(x) = 1 −qx. The app op ia e o mulae o he q-Cha lie
polynomials (31) a e he e o e easy consequences o he co esponding o mulae o he q-Meixne
polynomials (28) wi h he anishing alue o he pa ame e b.
16

4.2 Askey-Wilson cases
Using he linea i y o he di e ence equa ion (1) we can conside , wi h no loss o gene ali y,
he ollowing la ice x(s) = 1
2(qs+q−s), o which µ= 0. Then,
σ(s) = Cq−2s
4
Y
i=1
(qs−qsi) = q−2s
4
Y
i=1
(qs−zi), σ(−s−µ) = Cq2s
4
Y
i=1
(q−s−zi).
I is well known ha he gene al case when he ze os o ˜σ, namely z1z2z3z46= 0, co esponds
o he Askey-Wilson polynomials [7]. We will use he heo em 3.4 o sol e he ac o iza ion
p oblem o he whole q-Askey ableau [7] and Niki o o -U a o ableau [4, 5].
The Askey-Wilson polynomials on he la ice x(s) = 1
2(qs+q−s) = cos θwhe e qs=eiθ,
de ined by [7]
pn(x(s); a, b, c, d|q) = (ab, ac, ad;q)n
an4ϕ3 q−n, qn−1abcd, aq−s, aqs
ab, ac, ad q;q
!,
whe e a=z1,b=z2,c=z3,d=z4and µ= 0. Thei o hogonali y ela ion is o he o m
Z1
−1
ω(x)pn(x;a, b, c, d)pm(x;a, b, c, d)dx =δnmd2
n,
whe e
ω(x) = h(x, 1)h(x, −1)h(x, q 1
2)h(x, −q1
2)
2π√1−x2h(x, a)h(x, b)h(x, c)h(x, d), h(x, α) = ∞
Y
k=0
[1 −2αxqk+α2q2k],
and he no m is gi en by
d2
n=(abcdqn−1, abcdq2n;q)∞
(qn+1, abqn, acqn, adqn, bcqn, bdqn, cdqn;q)∞
.
The Askey-Wilson unc ions can be de ined by
Φn(s;q) = sω(s)A2(s)
d2
n
pn(x(s); a, b, c, d), x(s) = cos θ, qs=eiθ.
Taking A(s) = p∇x1(s), we ha e he o hogonali y R1
−1Φn(s;q)Φm(s;q)/∇x1(s)dx =δn,m,
and H(s;q)Φn(s;q) = λnΦn(s;q), whe e λn=q(q−n−1)(1 −abcdqn−1), and H(s;q) is gi en by
(9), σ(s) = Cσq−2s(qs−a)(qs−b)(qs−c)(qs−d) and µ= 0. Thus he Hamil onian, associa ed
wi h hese Askey-Wilson unc ions, is
H(s;q)= −1
k2
q√sin θ pσ(s)σ(−s+ 1)
sin(θ+i
2log q)psin(θ+ilog q)e−∂s+pσ(s+ 1)σ(−s)
sin(θ−i
2log q)psin(θ−ilog q)e∂s
!
+1
k2
qsin θσ(s)
sin(θ+i
2log q)+σ(−s)
sin(θ−i
2log q)I.
I we now use he Rema k 3.6, hen he i s condi ion o he Theo em 3.4 holds o α=1
2.
In ac , a s aigh o wa d calcula ions shows ha
∆x1(s+γ) = kq
2q−s−γ(qs+γ+ 1)(qs+γ−1).
17
Hence, he condi ion (18) has he o m (eiθ =qs)
q−2α+1 
qs−1
2−1
qs−α−1s(qs−1−1)(qs−1)
(qs−α−1
2−1)(qs−α+1
2−1)
u
u
4
Y
i=1
(qs−α−zi)(q−s+α−zi)
(qs−zi)(q−s+1 −zi)
×
qs−1
2+ 1
qs−α+ 1s(qs−1+ 1)(qs+ 1)
(qs−α−1
2+ 1)(qs−α+1
2+ 1) =ς.
I we look a he exp ession in he second line, we ind ha i is a cons an i and only i α=1
2,
and hus he i s condi ion ans o ms in o
sQ4
i=1(qs−1/2−zi)(q−s+1/2−zi)
Q4
i=1(qs−zi)(q−s+1 −zi)=ς. (32)
Then, he simples case o which he condi ion (18) holds is when z1z2=q1
2and z3z4=q1
2
(wi h he co esponding pe mu a ions o he oo s). In his case ς= 1. Since σ(s) = σ(−s+1
2),
he second condi ion gi es
1
∇x1(s) σ(s+1
2)
∆x(s)+σ(−s+1
2)
∇x(s)−σ(s)
∇x(s)−σ(−s)
∆x(s)!= 0.
Thus, we ha e a↑
1/2(s;q)a↓
1/2(s) = H(s;q) and [a↓
1/2(s;q), a↑
1/2(s)]q= 0, whe e
a↓
1/2(s;q) = e1
2∂ssσ(s)
−k2
qsin θsin(θ+i
2log q)−e−1
2∂ssσ(−s)
−k2
qsin θsin(θ−i
2log q),
a↑
1/2(s;q) = sσ(s)
−k2
qsin θsin(θ+i
2log q)e−1
2∂s−sσ(−s)
−k2
qsin θsin(θ−i
2log q)e1
2∂s.
Since he α-ope a o s a e commu ing, his case is no so in e es ing in applica ions (e.g., o
q-models o he ha monic oscilla o s). A special case o he Askey-Wilson polynomials a e
he con inuous q-Jacobi polynomials co esponding o he oo s a=qα0/2+1/4,b=qα0/2+3/4,
c=−qβ0/2+1/4,d=−qβ0/2+3/4[7], hen we can sol e he p oblem 1 only in he case when
α0=β0=−1/2.
The nex case is when one o he oo s zi anishes. This is he 0-Askey-Wilson polynomials
o he con inuous dual q-Hahn [7]. In his case he i s condi ion (18) (equi alen ly (32)) holds
only when
(z1, z2, z3) = ( , 1
2− , 1
4), ∈R.
Wi h he abo e choice o zii is impossible o ul ill he second condi ion (19) hence i is
impossible o ob ain a simple closed dynamical algeb a.
4.2.1 Con inuous q-Lague e polynomials
Le now conside he case when he Askey-Wilson polynomials ha e wo pa ame e s equal o
ze o. In o de ha he condi ion (18) ake place, he o he wo non- anishing pa ame e s should
sa is y ha hei p oduc is equal q1
2. Unde his condi ion ς=q−1
2. Then he second condi ion
yields Λ = 4Cσ(√q−1)
k2
q. Then, he Askey-Wilson Hamil onian wi h wo ze o oo s o σadmi s a
ac o iza ion wi h a non- i ial dynamical algeb a. An example o his amily is he con inuous
18
q-Lague e polynomials P(a)
n(x|q) [7], x(s) = cos θ, when a=−1/2 o which he Hamil onian
has he o m (A(s) = 1)
H(s;q)=−Cσ
kqsin θ
q(qs+1 −1)(qs+1 −q1
2)(q−s−1)(q−s−q1
2)
sin(θ+i
2log q)e−∂s+
+q(qs−1)(qs−q1
2)(q1−s−1)(q1−s−q1
2)
sin(θ−i
2log q)e∂s
+
−Cσ
kqsin θ (qs−1)(qs−q1
2)
sin(θ+i
2log q)+(q−s−1)(q−s−q1
2)
sin(θ−i
2log q)!I,
and
a↓
1/2(s;q) = √−Cσ
kqsin θe1
2∂sq(qs−1)(qs−q1
2)−e−1
2∂sq(q−s−1)(q−s−q1
2),
a↑
1/2(s;q) = √−Cσ
kqsin θq(qs−1)(qs−q1
2)e−1
2∂s−q(q−s−1)(q−s−q1
2)e1
2∂s.
Then o he unc ions
Φn(x(s)) = s(q1
2;q)n(q, q 1
2;q)∞ω(s)
(q;q)n3ϕ2
q−n, q−s, qs
q1
2,0q;q

, qs=eiθ, x(s) = cos θ,
we ha e he o hogonali y R1
−1Φn(x;q)Φm(x;q)dx =δn,m, and
H(s;q)Φn(s;q) = q(q−n−1)Φn(s;q),H(s;q) = a↑
1/2(s;q)a↓
1/2(s),
and
[a↓
1/2(s;q), a↑
1/2(s)]q−1/2=4Cσ(√q−1)
k2
q
,
Thus choosing Cσ=−k2
q
4(1−√q)we ob ain he ela ion [a↓
1/2(s;q), a↑
1/2(s)]q−1/2=I.
The case o Askey-Wilson polynomials wi h h ee ze o pa ame e s is analogue o he case o
one ze o pa ame e and he e is no possible o sol e he p oblem 1. An example o his case
a e he con inuous big q-He mi e polynomials [7].
4.2.2 Con inuous q-He mi e polynomials
Finally, i one akes he Askey-Wilson polynomials wi h anishing pa ame e s a, b, c, d, his gi es
he con inuous q-He mi e polynomials [7]. In his case σ(z) = Cσq2z.
Le choose A(s) = p∇x1(s). Taking in o accoun ha his amily is a special case o he
Askey-Wilson polynomials when all pa ame e s a=b=c=d= 0, one di ec ly ob ains, by
using ς= 1/q, ha in his case he Hamil onian is gi en by
H(s;q)= −1
k2
q√sin θ Cσq
sin(θ+i
2log q)psin(θ+ilog q)e−∂s+Cσq
sin(θ−i
2log q)psin(θ−ilog q)e∂s!
+1
k2
qsin θCσq2s
sin(θ+i
2log q)+Cσq−2s
sin(θ−i
2log q)I,
19
and he α-ope a o s
a↓
1/2(s;q) = e1
2∂ssCσq2s
−k2
qsin θsin(θ+i
2log q)−e−1
2∂ssCσq−2s
−k2
qsin θsin(θ−i
2log q)
a↑
1/2(s;q) = sCσq2s
−k2
qsin θsin(θ+i
2log q)e−1
2∂s−sCσq−2s
−k2
qsin θsin(θ−i
2log q)e1
2∂s
,
a e such ha
a↑(s;q)a↓(s;q) = H(s;q) and [a↓(s;q), a↑(s;q)]1/q =4Cσ
kq
.
No ice ha o ge ing he no malized commu a ion ela ions i is su icien o choose Cσ=kq/4.
Ano he possible choice is A(s) = 1 [12], hence a s aigh o wa d calculus shows ha he
wo condi ions in Theo em 3.4 a e ue i ς=q−1, hus Λ = 4Cσk−1
q. Wi h his choice he
o hogonali y o he unc ions Φnis R1
−1Φn(s;q)Φm(s;q)dx =δn,m. In his case, he Hamil onian
is equal o
H(s;q) = Cσq
k2
qe−∂s
sin θsin(θ+i
2ln q)+e∂s
sin(θ−i
2ln q) sin θ−4
√q1−1 + q
q+q−1−2 cos 2θI,
and
a↓(s;q) := a↓
1/2(s) = √−Cσ
kqsin θe1
2∂sqs−e−1
2∂sq−s,
a↑(s;q) := a↑
1/2(s) = √−Cσ
kqsin θqse−1
2∂s−q−se1
2∂s.
In e ms o hese ope a o s
H(s;q) = a↑(s;q)a↓(s;q) and [a↓(s;q), a↑(s;q)]q−1=4Cσ
kq
.
As be o e we now can choose Cσ=kq/4. This case was i s conside ed in [13], see also [25].
Appendix
He e we p esen a simple p oo o he Theo em 3.5 when α= 0. No ice ha in his case
b(s;q) = a†(s;q), i.e., he ope a o bis he adjoin o a.
Theo em 3.5a: Le {Φn(s;q)}be he eigen unc ions o he ope a o H(s;q), co esponding o
he eigen alues {λn(q)}. Suppose ha he H(s;q) admi s he ac o iza ion (13). I he ope -
a o s a(s;q) and b(s;q) in (13) sa is y he q-commu a ion ela ion [a(s;q), b(s;q)]q=I, hen
he eigen alues λn(q) a e q-linea o q−1-linea unc ions o n, i.e., ei he λn(q) = C1qn+C3o
λn(q) = C2q−n+C3, espec i ely.
P oo : Ob iously, he ope a o H(s;q) is diagonal in he basis consis ing o i s eigen unc ions
Φn(s;q). By hypo hesis, his ope a o admi s a he same ime he ac o iza ion (13) in e ms
o a(s;q) and b(s;q). Bu acco ding o p oposi ion 3.2 he unc ion a(s;q)Φn(s;q) is also he
eigen unc ion o H(s;q), associa ed wi h he eigen alue q−1[λn(q)−1]. Hence he a(s;q) is ei he
he lowe ing ope a o o he aising ope a o . In he o me case his means ha
q−1[λn(q)−1] = λn−1(q) + C ,
20
om which i ollows ha C2= 0 (i.e., he spec um {λn(q)}is a q-linea one) and C=
q−1[(1 −q)C3−1]. In la e case he co esponding ela ion is
q−1[λn(q)−1] = λn+1(q) + C ,
which holds when C1= 0 (i.e., he spec um is q−1-linea ) and C=q−1[(1 −q)C3−1]. The
p oo o he heo em is hus comple e.
Acknowledgmen s: The esea ch o RAN has been pa ially suppo ed by he Minis e io de
Ciencias y Tecnolog´ıa o Spain unde he g an BFM 2003-06335-C03-01, he Jun a de Andaluc´ıa
unde g an FQM-262. The pa icipa ion o NMA in his wo k has been suppo ed in pa by he
UNAM–DGAPA p ojec IN102603-3 “ ´
Op ica Ma em´a ica” and he Jun a de Andaluc´ıa unde
g an 2003/2. Finally, we a e g a e ul o bo h he e e ees o hei ema ks and sugges ions,
which helped us o imp o e he exposi ion o ou esul s.
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