On a q-ex ension o he linea ha monic oscilla o
wi h he con inuous o hogonali y p ope y on R
R. ´
Al a ez-Noda se∗, M. K. A akishiye a†, and N. M. A akishiye ‡
∗Depa amen o de An´alisis Ma em´a ico, Uni e sidad de Se illa,
Apdo. 1160, E-41080 Se illa, Spain and
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
E-mail: [email p o ec ed]
†Facul ad de Ciencias, UAEM, Apa ado Pos al 396-3,
CP 62250, Cue na aca, Mo elos, M´exico
E-mail: [email p o ec ed]
‡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
E-mail: na i[email p o ec ed]
Abs ac
We discuss a q-analogue o he linea ha monic oscilla o in quan um mechanics, based
on a q-ex ension o he classical He mi e polynomials Hn(x), ecen ly in oduced by us
in [1]. The wa e unc ions in his q-model o he quan um ha monic oscilla o possess
he con inuous o hogonali y p ope y on he whole eal line Rwi h espec o a posi i e
weigh unc ion. A de ailed desc ip ion o he co esponding q-sys em is ca ied ou .
1 In oduc ion
In [1] we in oduced a q-ex ension o he classical He mi e polynomials Hn(x), which sa is y
he ollowing equi emen s: They a e polynomials in he a iable x, which obey a h ee- e m
ecu ence ela ion; They a e o hogonal on he whole eal line Rwi h espec o a con inuous
posi i e weigh unc ion; In he limi as q→1 hey coincide wi h he He mi e polynomials
Hn(x). Such a amily enables one o build a q-de o med e sion o he linea ha monic
oscilla o in quan um mechanics, which is s ill de ined on he whole eal line Rand enjoys
he con inuous o hogonali y p ope y on Rwi h espec o a posi i e weigh unc ion. Le us
poin ou he e ha he e a e se e al publica ions (see [2]–[10] and e e ences he ein) de o ed
o he s udy o explici ealiza ions, which ep esen q-ex ensions o he He mi e unc ions
(o he wa e unc ions o he linea ha monic oscilla o ) Hn(x)e−x2/2. Bu none o hese
ealiza ions sa is ies all o he a o emen ioned equi emen s: he con inuous weigh unc ions
in [2, 4, 7] a e suppo ed on he ini e in e als; he con inuous weigh unc ions in [3, 8] a e
no posi i e; he q-ex ensions in [2], [4]–[9] a e no exp essed in e ms o polynomials in he
independen a iable; and, inally, he o hogonali y ela ions in [5]–[7], [10] a e disc e e.
Ou main goal in his pape has been o employ his q-ex ension o he He mi e polyno-
mials, Hn(x;q), in o de o buil a q-analogue o he linea ha monic oscilla o in quan um
mechanics. Sec ion 2 collec s hose known esul s om [1] abou he polynomials Hn(x;q),
which a e needed in sec ion 3 o de i e an explici o m o he wa e unc ions ψn(x;q) in his
q-model and hei p ope ies. Sec ion 4 is de o ed o explici cons uc ion o he gene a o s
o he dynamical symme y algeb a suq(1,1) in e ms o he lowe ing and aising q-di e ence
ope a o s a(x;q) and a†(x;q). Concluding sec ion 5 con ains a b ie discussion o q-cohe en
s a es o his q-ex ension o he quan um ha monic oscilla o .
1
2 De ini ion and p ope ies o he polynomials Hn(x;q)
In [1] he ollowing amily was in oduced
H2n(x;q) := (−1)n(q;q)nL(−1/2)
n(x2;q)
= (−1)n(q1/2;q)n1φ1 q−n
q1/2
q;−qn+1/2x2!= (−1)n2φ1 q−n,−x2
0
q;qn+1/2!,
H2n+1(x;q) := (−1)n(q;q)nx L(1/2)
n(x2;q)
= (−1)n(q3/2;q)nx1φ1 q−n
q3/2
q;−qn+3/2x2!= (−1)nx2φ1 q−n,−x2
0
q;qn+3/2!,
(2.1)
whe e L(α)
n(x;q) a e q-Lague e polynomials, 1φ1and 2φ1deno e he basic hype geome ic
polynomials and (a;q)nis he q-shi ed ac o ial (we employ s anda d no a ions o q-analysis,
see, o example, [11] o [12]). In (2.1) and h oughou he sequel i is assumed ha qis a
ixed numbe such ha 0 < q < 1.
This amily is gene a ed by he h ee- e m ecu ence ela ion
xHn(x;q) = q−n/2Hn+1(x;q)−(1 −q−n/2)Hn−1(x;q), n = 0,1,2, ... , (2.2)
wi h he ini ial condi ion H0(x;q)≡1.
The polynomials (2.1) sa is y he con inuous o hogonali y ela ion
∞
Z
−∞ Hm(x;q)Hn(x;q)dx
Eq(x2)=πq−n/2(q1/2;q1/2)n(q1/2;q)1/2δmn (2.3)
on he whole eal line Rwi h espec o he posi i e weigh unc ion w(x) = 1/Eq(x2) =
1/(−x2;q)∞[1].
The polynomials Hn(x;q) cons i u e a q-ex ension o he classical He mi e polynomials
Hn(x) since hese polynomials educe o he la e in he limi as q→1 , i.e.,
lim
q→1(1 −q)−n/2Hn(p1−q x;q) = 2−nHn(x),(2.4)
F om he ecu ence ela ion (2.2) i ollows ha he Hn(x;q) can be exp essed in e ms
o he disc e e q-He mi e polynomials ˜
h(x;q) o ype II as
Hn(x;q2) = qn(n−1)/2˜
hn(x;q) := i−n2φ0 q−n, ix
−
q;−qn!.(2.5)
So om he known q-di e ence equa ion o he disc e e q-He mi e polynomials ˜
hn(x;q) (see
[13], (3.29.5), p.119) one deduces ha
(1 −qn/2)x2Hn(x;q) = (1 + q1/2+x2)Hn(x;q)
−(1 + x2)Hn(q1/2x;q)−q1/2Hn(q−1/2x;q).
(2.6)
Simila ly, one eadily e i ies ha he o wa d and backwa d shi ope a o s o he polyno-
mials Hn(x;q) a e o he o m
hq−1
2xd
dx −1iHn(x;q) = q−1/2(1 −qn/2)xHn−1(x;q),
h(1 + x2)q1
2xd
dx −1iHn(x;q) = xHn+1(x;q),
(2.7)
2
espec i ely, whe e qa x d
dx is he dila ion ope a o , i.e., qa x d
dx (x) = (qax).
A Rod igues- ype di e ence o mula o he polynomials Hn(x;q) can be w i en as
Hn(x;q) = (−x)−nEq(x2) (q1
2xd
dx ;q−1/2)nE−1
q(x2),(2.8)
whe e we ha e sligh ly simpli ied he n- h powe o he q-de i a i e ope a o Dq(c (3.29.10)
in [13], p.119) by ep esen ing i in he o m
Dn
q≡1
(1 −q)nxn(qxd
dx ;q−1)n, n = 0,1,2, ... . (2.9)
I is no di icul o p o e (2.9) by induc ion on he powe n.
Finally, using he gene a ion unc ion o he disc e e q-He mi e polynomials ˜
hn(x;q) o
ype II [13], one inds ha
(−x ;q1/2)∞
(− 2;q)∞
=∞
X
n=0
1
(q1/2;q1/2)nHn(x;q) n.(2.10)
3 Wa e unc ions ψn(x;q)and hei p ope ies
We wish o discuss a q-model o he linea ha monic oscilla o , which is desc ibed by he
wa e unc ions o he o m
ψn(x;q) := d−1
n(q)Hn(x;q)E−1/2
q(x2) (3.1)
wi h he no maliza ion cons an dn(q) := q−n/4qπ(q1/2;q)1/2(q1/2;q1/2)n. Then, by con i-
nuous o hogonali y ela ion (2.3), hese unc ions a e o hono mal on R, ha is,
∞
Z
−∞
ψm(x;q)ψn(x;q)dx =δmn.(3.2)
The wa e unc ions ψn(x;q) a e de ined by (3.1) in such a way ha in he limi as q→1
hey coincide wi h he o hono malized He mi e unc ions (o he wa e unc ions o he linea
ha monic oscilla o in non- ela i is ic quan um mechanics):
lim
q→1ψnp1−q ξ;q=1
p√π2nn!Hn(ξ) exp (−ξ2/2) =: ψn(ξ).(3.3)
This limi p ope y o ψn(x;q) ollows immedia ely om (2.4) and he well-known ac
lim
q→1Eq((1 −q)z) = ez(3.4)
abou he Jackson q-exponen ial unc ion Eq(z) (see [11] o [12]).
F om (2.6) and (3.1) one ob ains ha he wa e unc ions ψn(x;q) a e eigen unc ions o
he q-Hamil onian H(x;q),
H(x;q)ψn(x;q) = En(q)ψn(x;q), En(q) := 1−qn/2
1−q1/2.(3.5)
By equa ion (2.6), he explici o m o his sel -adjoin q-di e ence ope a o is
H(x;q) := 1
(1−q1/2)x2h(1 + x2+q1/2)I−p1 + x2q1
2xd
dx −q1
2(1−xd
dx )p1 + x2i,(3.6)
3
whe e Iis he iden i y ope a o . This exp ession o H(x;q) in e ms o he dila ion ope a o s
q±1
2xd
dx may c ea e an imp ession ha he H(x;q) con ains singula i y a x= 0 due o he
p esence o he ac o x2in he denomina o . To emo e his doub one should ake in o
accoun ha , by de ini ion (3.6),
H(x;q)ψn(x;q) = 1
(1−√q)x2(1 + √q+x2)ψn(x;q)
−p1 + x2ψn(q1/2x;q)−pq+x2ψn(q−1/2x;q)i(3.7)
o all n= 0,1,2, ... . Besides, om (3.1) i is e iden ha he wa e unc ions ψn(x;q) ha e
egula beha io a ound x= 0. Now subs i u ing he sum o i s wo e ms c0+c1x om he
expansion o ψn(x;q) a ound x= 0 in o exp ession in squa e b acke s in (3.7) and keeping
only cons an and linea in x e ms, one eadily e i ies ha
(1 + √q) (c0+c1x)−(c0+c1√q x)−√qc0+c1
√qx= 0 .
Consequen ly, he o al combina ion inside he squa e b acke s in (3.7) beha es like x2in he
x→0 limi and he igh side o (3.7) he e o e assumes a cons an alue a x= 0. This
con i ms ha he e is no singula i y a x= 0.
We obse e also ha he eigen alues En(q) o H(x;q) a e bounded om abo e by he
asymp o ic alue E∞(q) = 1/(1 −q1/2) and, since En+1(q)−En(q) = qn/2, hey a e no
equidis an .
F om (2.2) i ollows ha he wa e unc ions ψn(x;q) sa is y he h ee- e m ecu ence
ela ion
x ψn(x;q) = q−(2n+1)/4q1−q(n+1)/2ψn+1(x;q) + q(1−2n)/4q1−qn/2ψn−1(x;q) (3.8)
wi h he ini ial condi ion ha he g ound s a e ψ0(x;q) = d−1
0(q)E−1/2
q(x2).
Likewise, om he explici o m o he o wa d and backwa d shi ope a o s (2.7) i
ollows ha
a(x;q)ψn(x;q) = pEn(q)ψn−1(x;q), a†(x;q)ψn(x;q) = pEn+1(q)ψn+1(x;q),(3.9)
whe e he q-di e ence lowe ing and aising ope a o s a(x;q) and a†(x;q) a e gi en by
a(x;q) = q1/4
√1−q1/2xq−1
2xd
dx √1 + x2−I,
a†(x;q) = q1/4
√1−q1/2x√1 + x2q1
2xd
dx −I,
(3.10)
espec i ely. We in i e he eade o e i y ha hese ope a o s a e indeed mu ually adjoin
in he Hilbe space L2(R, dx) o squa e in eg able unc ions (x) wi h espec o dx.
Simila o he case o he quan um linea ha monic oscilla o , he lowe ing and aising
ope a o s (3.10) ac o ize he Hamil onian (3.6), ha is,
H(x;q) = a†(x;q)a(x;q).(3.11)
Mo eo e , i is no di icul o e i y, by using (3.10), ha hei ano he (i.e., when he
ope a o a(x;q) is igh mul iplied by i s adjoin ope a o a†(x;q)) p oduc a(x;q)a†(x;q)
4
is equal o I+q1/2H(x;q). This means ha he ope a o s a(x;q) and a†(x;q) sa is y he
q-commu a ion ela ion o he o m
a(x;q)a†(x;q)−q1/2a†(x;q)a(x;q)≡ha(x;q), a†(x;q)iq1/2=I . (3.12)
I should be no ed a his poin ha we ha e used abo e he known explici o m o he
o wa d and backwa d shi ope a o s (2.7) o he polynomials Hn(x;q) in o de o ind he
lowe ing and aising ope a o s a(x;q) and a†(x;q). Bu we could ha e s a ed equi alen ly
wi h he q-di e ence equa ion (3.5) i sel and ha e di ec ly ac o ized i in e ms o he same
ope a o s a(x;q) and a†(x;q) ( o a mo e de ailed discussion o he ac o iza ion o di e ence
equa ions, see, o example, [15, 16]).
So we ha e es ablished ha ou q-model is go e ned by he Hamil onian (3.6), which
admi s he ac o iza ion (3.11) in e ms o he ope a o s a(x;q) and a†(x;q), sa is ying he q-
commu a ion ela ion (3.12). This cha ac e is ic p ope y o he Hamil onian (3.6) is known
o e lec he ac ha he dynamical symme y o his q-model is desc ibed by he quan um
algeb a suq(1,1) [14]. In he nex sec ion we cons uc explici ly he gene a o s o his algeb a
in e ms o he lowe ing and aising ope a o s a(x;q) and a†(x;q).
4 Dynamical symme y
In his sec ion we emind he eade i s how one cons uc s a dynamical symme y algeb a
o he linea ha monic oscilla o , which is go e ned in non- ela i is ic quan um mechanics
by he well-known Hamil onian
H(x) := ~ω
2ξ2−d2
dξ2≡~ωN(x) + 1
2,(4.1)
whe e ξ=pmw/~xis a dimensionless coo dina e, N(x) is he pa icle numbe ope a o ,
N(x) := a†(x)a(x),(4.2)
and he annihila ion and c ea ion ope a o s a e de ined as usual:
a(x) = 1
√2ξ+d
dξ , a†(x) = 1
√2ξ−d
dξ ,
a(x), a†(x)≡a(x)a†(x)−a†(x)a(x) = I .
(4.3)
By using (4.1) and (4.3) one eadily e i ies ha
[H(x), a(x) ] = −a(x),hH(x), a†(x)i=a†(x).(4.4)
Obse e ha in he case o he linea ha monic oscilla o (4.1) he e is no much di e ence
be ween he Hamil onian H(x) and he pa icle numbe ope a o N(x): he o me ope a o ,
di ided by he ac o ~ω, is equal o he la e one plus a cons an e m 1/2. So, he pa icle
numbe ope a o N(x) sa is ies he same commu a ion ela ions (4.4) wi h he annihila ion
and c ea ion ope a o s a(x) and a†(x).
Ha ing ac o ized he Hamil onian H(x) (o , equi alen ly, he pa icle numbe ope a o
N(x)) in e ms o he annihila ion a(x) and c ea ion a†(x) ope a o s, one explici ly cons uc s
he closed Lie algeb a su(1,1) wi h he h ee gene a o s
K0(x) := 1
2~ωH(x)≡1
2N(x) + 1
2, K+(x) := 1
2a†(x)2, K−(x) := 1
2a2(x).(4.5)
5

Indeed, i is no di icul 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(x), K±(x)] = ±K±(x),[K−(x), K+(x)] = 2 K0(x),(4.6)
o he algeb a su(1,1). Uni a y i educible ep esen a ions o his algeb a a e known o be
cha ac e ized by eigen alues o he in a ian ( ha is, commu ing wi h all h ee gene a o s
(4.5)) Casimi ope a o
C:= K0(x) [ K0(x)−I]−K+(x)K−(x) = s(s−1) I . (4.7)
A di ec calcula ion o he Casimi ope a o (4.7) wi h he aid o (4.5) shows ha he
eigen alue s(s−1) in his pa icula case is equal o −3/16. This means ha he pa ame e
smay be equal o ei he s1= 1/4 o s2= 3/4. Each o hese wo alues o sde ines a uni a y
i educible ep esen a ion o he algeb a su(1,1): D+(1/4) consis s o hose eigens a es o
he Hamil onian H(x), which co espond o he eigen alues s1+n=n+1/4 = (2n+1/2)/2,
n= 0,1,2, ..., o he gene a o K0(x) = H(x)/2~ω; whe eas D+(3/4) co esponds o he
eigen alues s2+n=n+ 3/4 = (2n+ 1 + 1/2)/2 o he same gene a o K0(x). So in his way
one a i es a he co ec spec um En=~ω(n+ 1/2) o he Hamil onian H(x), wi hou
sol ing an eigen alue p oblem o he app op ia e Sch ¨odinge equa ion. Thus eigens a es
o H(x) wi h he eigen alues E2n o m he uni a y i educible ep esen a ion D+(1/4) and
hose wi h E2n+1 o m ano he one, D+(3/4).
The Fock space HFo all eigen unc ions {ψn(x)}o he Hamil onian H(x) spli s in o wo
su(1,1)-i educible subspaces o H(x) is symme ic wi h espec o he in e sion x→ −x.
The e o e he in e sion ope a o P,P x =−x, commu es wi h all h ee gene a o s (4.5) and
HFdecomposes in o wo i educible componen s,
HF=H0⊕ H1,(4.8)
consis ing o he wa e unc ions ψn(x) wi h e en and odd indices n, espec i ely. The i e-
ducible subspaces H0and H1a e cha ac e ized by he eigen alues (−1)ǫo he ope a o P
wi h ǫ= 0 in H0and ǫ= 1 in H1. I is clea ha he subspaces H0and H1co espond o
he uni a y i educible ep esen a ions D+(1/4) and D+(3/4), espec i ely.
Now we a e in a posi ion o discuss a dynamical symme y algeb a o he q-model (3.1).
To cons uc i one needs o in oduce i s he ope a o [14]
N(x;q) := 2
ln qln h1−(1 −q1/2)H(x;q)i.(4.9)
Since he wa e unc ions ψn(x;q) a e eigen unc ions o he q-Hamil onian Hn(x;q) wi h he
eigen alues En= (1 −qn/2)/(1 −q1/2), om he de ini ion (4.9) one deduces ha
N(x;q)ψn(x;q) = n ψn(x;q),(4.10)
ha is, N(x;q) is he pa icle numbe ope a o and
[N(x;q), a(x;q)] = −a(x;q),hN(x;q), a†(x;q)i=a†(x;q).(4.11)
A he nex s ep one de ines a new se o he ope a o s
b(x;q) := q−N(x;q)/8a(x;q), b†(x;q) := a†(x;q)q−N(x;q)/8,(4.12)
which sa is y, acco ding o (4.11), he ollowing commu a ion ela ion
b(x;q)b†(x;q)−q1/4b†(x;q)b(x;q) = q−N(x;q)/4.(4.13)
6
This is eadily e i ied wi h he aid o (4.11). The ope a o s b(x;q), b†(x;q), and N(x;q)
di ec ly lead o he dynamical algeb a suq1/2(1,1) wi h he gene a o s
K+(x;q) := γb†(x;q)2, K−(x;q) := γ b2(x;q), K0(x;q) := 1
2N(x;q) + 1
2,
γ= [ 1/2 ]q1/2.
(4.14)
I is no di icul o check ha hus de ined gene a o s (4.14) sa is y he s anda d com-
mu a ion ela ions
[K0(x;q), K±(x;q)] = ±K±(x;q),[K−(x;q), K+(x;q)] = [2K0(x;q)]q1/2,(4.15)
o he quan um algeb a suq1/2(1,1). The q-numbe [ A]qin (4.14) is gi en by he common
exp ession
[A]q:= qA−q−A
q−q−1.(4.16)
We a e in e es ed in he posi i e disc e e se ies ep esen a ions o he quan um algeb a
suq(1,1) wi h lowes weigh s. These i educible ep esen a ions o suq(1,1) a e deno ed by
T+
l, whe e lis he lowes weigh , which can be any posi i e numbe (see, o example, [17]).
I is he cha ac e is ic p ope y o e e y T+
l ha he gene a o K0(x;q) has he eigen alues
l+n,n= 0,1,2, ..., in T+
l.
The in a ian Casimi ope a o in he case unde discussion is equal o
C(q) := [ K0(x;q)−1/2 ]2
q1/2−K+(x;q)K−(x;q)−1
4I=[ 1/4 ]2
q1/2−1/4I. (4.17)
This means ha wo possible alues o he pa ame e sin his case a e
s1(q) = 1/2−[ 1/4 ]q1/2, s2(q) = 1/2 + [ 1/4 ]q1/2.(4.18)
Since [a]q→ain he limi as q→1 by de ini ion o he q-numbe (4.16), he eigen alue o
C(q) in (4.17) educes in his limi o he eigen alue o he Casimi ope a o in he case o
he linea ha monic oscilla o (4.7). E iden ly, he same happens wi h he alues o s1(q)
and s2(q): hey coincide in his limi wi h he co esponding alues o he pa ame e sin
(4.7), i.e.,
lim
q→1s1(q) = 1
4,lim
q→1s2(q) = 3
4.(4.19)
F om (4.17) i now ollows ha he lowes weigh s in ou case a e 1/4 and 3/4. The e o e
by (4.14) he eigen alues o he pa icle numbe ope a o N(x;q)≡2K0(x;q)−1/2 a e equal
o 2nand 2n+1, n= 0,1,2, ..., espec i ely. Taking in o accoun in e ela ion (4.9) be ween
he ope a o s N(x;q) and H(x;q), one hus a i es a he co ec spec um (3.5) o he
Hamil onian H(x;q), wi hou sol ing an eigen alue p oblem o H(x;q).
So we conclude ha he wa e unc ions ψn(x;q), de ined in (3.1), o m a ep esen a ion
o he quan um algeb a suq1/2(1,1) in he Fock space HF. This ep esen a ion in he space
HFis educible p ecisely o he same eason as in he case o he linea ha monic oscilla o
(4.1). Thus HFspli s in o wo suq1/2(1,1)-i educible subspaces H0≡T+
1/4and H1≡T+
3/4,
consis ing o he wa e unc ions ψn(x;q) wi h e en and odd indices n, espec i ely.
7
5q-cohe en s a es
As in he case o he non- ela i is ic linea ha monic oscilla o , one can cons uc q-cohe en
s a es o his model as eigen unc ions o he lowe ing ope a o a(x;q), ha is,
a(x;q)ϕζ(x;q) = ζ ϕζ(x;q),(5.1)
whe e ζis some a bi a y numbe . To ind an explici o m o hese s a es ϕζ(x;q), we i s
no e ha by (3.8)
ψn(x;q) = cn(q)ha†(x;q)inψ0(x;q), cn(q) := s(1 −q1/2)n
(q1/2;q1/2)n
.(5.2)
Consequen ly, wi h he aid o (3.8) i is no di icul o e i y ha he s a es
ϕζ(x;q) := q(ζ)∞
X
n=0
cn(q)ζnψn(x;q),(5.3)
whe e q(ζ) is some no maliza ion ac o (see below), a e indeed he eigens a es o he ope-
a o a(x;q) wi h he eigen alues ζ. They o m an o e comple e sys em in he Hilbe space
HFand hey a e no o hogonal in his space. In ac , one can p o e, by using expansion
(5.3) and o hogonali y ela ion (3.2), ha
Z∞
−∞
ϕζ(x;q)ϕζ′(x;q)dx = q(ζ) q(ζ′)eq1/2(1 −q1/2)ζ ζ′,(5.4)
whe e
eq(z) := ∞
X
n=0
zn
(q;q)n
=1
(z;q)∞
,|z|<1.
The no maliza ion condi ion ha he in eg al on he le o (5.4) is equal o 1 equi es o
choose q(ζ) = qEq1/2−(1 −q1/2)ζ2. Thus,
ϕζ(x;q) = qEq1/2−(1 −q1/2)ζ2∞
X
n=0
cn(q)ζnψn(x;q),(5.5)
Subs i u e now in o expansion (5.5) explici o m o he coe icien s cn(q) om (5.2) and he
no maliza ion cons an s dn(q) o he wa e unc ions ψn(x;q) om (3.1) and employ hen
he gene a ing unc ion (2.10) o he polynomials Hn(x;q). This yields he inal o m o he
no malized q-cohe en eigen unc ions o he lowe ing ope a o a(x;q):
ϕζ(x;q) = sEq1/2−(1 −q1/2)ζ2
π(q1/2;q)1/2Eq(x2)
Eq1/2q1/4p1−q1/2xζ 
Eqq1/2(1 −q1/2)ζ2.(5.6)
Acknowledgmen s: The esea ch o RAN has been pa ially suppo ed by he DGES g an
BFM 2003-06335-C03-01 and PAI g an FQM-0262. The pa icipa ion o NMA in his wo k
has been suppo ed in pa by he UNAM–DGAPA g an IN102603-3 ´
Op ica Ma em´a ica.
The main pa o his wo k was pe o med du ing a isi by NMA o he Facul ad de
Ma em´a icas, Uni e sidad de Se illa, in June, 2004; he g a e ully acknowledges he suppo
o his isi by he Jun a Andaluc´ıa, g an 2003.
8
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9