- Email: [email protected]
Mutual integrability, quadratic algebras, and dynamical symmetry
ANNALS
OF PHYSICS
217,
1-20
Mutual
YA.
I.
(1992)
Integrability, Quadratic Algebras, and Dynamical Symmetry
GRANOVSKII, Physics
I. M.
LUTZENKO,
A. S. ZHEDANOV
AND
Department, Donetsk University, Donetsk, 340055, Ukraine
Received September 11, 1991
The concept of mutually integrable dynamical variables is proposed. This concept leads to the quadratic Askey-Wilson algebra QAW(3) which is the dynamical symmetry algebra for all problems where the most general “classical” polynoials arise. In classical mechanics the algebra of the same structure describes the time evolution of dynamical variables in terms of elementary functions. We apply the special case of QAW(3kJacobi algebra-to describe the dynamical symmetry of exactly solvable potentials and to resolve the “Manning mystery”-the intimate relation between classical and quantum exactly solvable potentials. :D 1992 Academic Press, Inc.
1. INTRODUCTION More than a half-century ago M. Manning observed [ 1] that for each exactly solvable quantum problem there exists an integrable classical analogue, and, moreover, it is integrable in terms of elementary functions. Why does this happen? The obvious simplicity of this question should not hide the deep problems standing behind it. First, it is a problem of the homomorphism between quantum and classical approaches to the same object. The primary answer to this question is the quasiclassical treatment of the lines of the Bohr-Sommerfeld quantization rules. Manning’s observation is related to the situation when quasiclassics are not sufficient-but does that exhaust the subject? And what is the extended principle of quantization? The generally accepted and simple requirement of quadratic integrability of the wave function works only for simple potentials (with only one singular point) and collapses for more complicated ones where the problem of monodromy arises. The second problem is tightly connected with the widespread dogma that both potentials-classical and quantum-are identical and U(r) only becomes an operator in quantum-mechanical treatment! Is this true, indeed, or may it be due to this transfer that the specific quantum corrections come up? Manning’s correspondence puts definite restrictions on the possible structures of potentials. GQO3-4916192$9.00 Copyright d 1992 by Academic Press, Inc. All rights of reproductmn in any lorm reserved.
2
GRANOVSKII,LUTZENKO,
AND ZHEDANOV
Third, we would like to have a constructive arrangement, such as a Feynman path integral or something else, that would realise the needed homomorphism. Unfortunately, the technical difficulties (only Gaussian integrals or only soliton-like potentials) bring these attempts again to the quasiclassical route. Fourth, and finally, there should exist an abstract scheme that branches to either the quantum or the classical version along the way of its realization. The obvious candidate might be the dynamical symmetry group because of its intimate connection between exact solvability and intrinsic symmetry. To guess the group structure that would embrace all of the wide spectrum of Manning’s problems is a very non-trivial job with low chances for success. Nevertheless, we present an attempt moving just along the lines of a grouptheoretical approach. It contains a further development of our papers [2,3 3 on dynamical algebras containing all the exactly soluble problems of quantum mechanics. In Section 2 we propose a concept of mutual integrability that allows us to construct the quadratic algebra QAW(3), describing the wide class of “exactly solvable” classical problems (i.e., the time dependence of variables is expressed in terms of elementary functions). In Section 3 a quantum analog of this algebra-QAW(3 jis introduced. This algebra describes all the problems where the most general Askey-Wilson polynomials (and their degenerations) arise as eigenfunctions. In Section 4 we concentrate on the representations of Jacobi algebra QJ(3) (which is the special case of QAW(3)). The representations mentioned are used in Sections 5 and 6 to describe all the exactly solvable potentials and to resolve the “Manning mystery.”
2. MUTUAL
INTEGRABILITY-~LASSICAL
QAW(3)
We introduce the concept of mutual integrability proceeding from the classical mechanics. Let ki be the family of dynamical variables defined on the phase space with Poisson brackets denoted by (., .). The time dynamics is defined by the standard Poisson equation ki = (H, kJ,
(2-l)
where H is the Hamiltonian. Now we concentrate our attention on the two dynamical variables denoted by k, and k2 and let k, be the Hamiltonian: H= k,. Let us demand that the equation for k2 has the form -it,
where F,(k,) and t&(k,)
= F,(k,) kz + @,(k,),
are arbitrary
functions of k,.
(2.2)
MUTUAL
3
INTEGRABILITY
Because k, = const it is clear that the solution of (2.2) is an elementary function of time, if if if
cl exp(ot) + c2 exp( -ot) + c3 cl exp(iot) + c2 exp( -jot) + c3
k,(t) =
1 c,t2+c2t+c,
F,-cO
(2.3)
F,>O
(2.4)
F, = 0,
(2.5)
where o(k,) = dm, c,-arbitrary functions of k,. One can write Eq. (2.2) in algebraic form. To this aim let us define the third dynamical variable k, as the Poisson bracket of k, and k,: G.6)
k,=(k,,k,)
As k, is the Hamiltonian
then I;,=k, (2.7)
i;, = k, = (k,, k3)
and Eq. (2.2) can be rewritten in the form (k,, k,)=FAk,)
k,+ @,(k,).
(2.8)
Now we define the property of “relative integrability.” Namely, we say that dynamical variable k, is integrable relative to k, if the relations (2.6) and (2.8) are taking place. Because of the arbitrariness of the functions F2 and Q2 this definition is valid for the wide class of dynamical variables. However, one can reduce this class if the “mutual” integrability takes place. We say that variables k, and k, are “mutually integrable” if they are simultaneously integrable, one relative to the other. The property of mutual integrability can be written in the algebraic form: (k,, kz)=k, (2.9)
(k,, k,) = Fl(kJ k, + @l(kd (k,, k, I= F,(k, 1 k, + @z(k, ).
Despite the fact that property of mutual integrability seems to be very simple, it gives rise to many non-trivial consequences concerning the dynamical symmetry of classical (and quantum) problems. First, it strictly constrains the function Fi, @, due to relations (2.9). The relations (2.9) define a nonlinear Poisson algebra of generators ki. These generators should obey the Jacobi identity: (k,(k,, k3)) + Mk,,
k,)) + Mk,,
W = 0.
(2.10)
Inserting (2.9) into (2.10) one obtains that Fi and Di should be, in general, the
4
GRANOVSKII,
polynomials variables),
LUTZENKO,
AND
ZHEDANOV
of the second order (we treat ki and k, as algebraically independent F&k,) = Zrk; + Za,k, + cl FJk,)
= 2rkf + 2azk, + c2
(2.11)
Ql(k2) = a,k; + dkz + g, @P,(k,) = a&: + dk, + g,,
wherer, a,,,, clv2, d, g,,, are arbitrary parameters. We can now write the defining relations for the algebra (2.9) in explicit form [4],
(k,, k,)=h (k2, k,)=2rkgk,
+2a,k,k2+azk~+r,k,+dk,+gl
(k3, k,) = 2rk:k,
+ al kf + 2a2k, k2 + c,kz + dk, + g,.
(2.12)
The relations (2.12) define the bi-quadratic algebra of generators ki. By “quadratic algebra” we mean the object being introduced by Sklyanin [S J: the Poisson brackets of the generators are expressible in terms of quadratic (and linear) combinations of the generators. In the case of Eqs. (2.12), the right-hand sides of the relations are quadratic relative to both the kl and k, generators. We see that the concept of “mutual integrability” is the most natural way for introducing quadratic algebra into the dynamics. However, the arguments for introducing the first examples of quadratic algebras (Sklyanin algebras) were quite different (see [5]). Moreover, the Sklyanin’s algebras have more complicated structure and in general do not obey the property of mutual integrability. It will be shown that quadratic algebras defined by (2.12) are playing the important role in almost all exactly solvable problems of classical and quantum mechanics. Now we propose an appropriate terminology for the different cases of algebra (2.12). If r # 0 we call this algebra QAW(3)-quadratic Askey-Wilson algebra with three generators (the meaning of this and the following notations will be clear in the next sections). If r = 0 but a, az # 0 we obtain the so-called quadratic Racah algebra Qr(3). If r = a1 = 0 the corresponding algebra is called Hahn algebra Qh(3). If r = a1 = c, = 0 one obtains the quadratic Jacobi algebra Qj(3). Finally, if al = a, = 0 one obtains the ordinary Lie algebra (with three generators) isomorphic to either su(2), su(1, l), or Heisenberg-Weyl (oscillator) algebra (see Table I). One can obtain the Casismir element q for the algebra (2.12): (q,k,)=(q,k,)=(q,k,)=O. Indeed, from Eq. (2.2) we have $ [k: t F,(k,) k; + 2Qj2(kl) k,] = 0
(2.13)
MUTUAL
5
INTEGRABILITY
TABLE
I
Nomenclature of Quadratic Algebras
QAW(3) QR(3) QW)
QJ(3) Lie
*
* *
0
t,
*
0
0
0
0
Fl
* *
ii ;
*
or
k: + FAk, ) k: + 2@,(k,) k, = W, ), where ((k,) is some function of k,. In analogous way, by choosing k, to be the Hamiltonian, relation
k: + F,(k) k: +2@,(h) k, = v(k,)
(2.14)
one obtains the (2.15)
with some function q(k,). Combining (2.14) and (2.15) one obtains the expression for the Casimir element q independent of time in both cases:
+c,k;+c,k:+2dk,k,+2g,kl+2g,k,.
(2.16)
It is not difficult to verify that expression (2.16) indeed yields the Casimir element-one needs to directly calculate the Poisson brackets (q, kJ. It is interesting to note that contrary to Lie algebras where the Casimir element is quadratic in the generators, our Casimir element q is wholly of the fourth order but quadratic separately in k,, k,, k,. In particular, if k, is the Hamiltonian, then k, = const and q defines the quadratic curve in the variables k, and k,. Because all the quadratic curves can be parametrized via the elementary functions we obtain one more interpretation of the mutual integrability. Indeed, the time t is the parameter providing the description of k2 in terms of elementary functions. An analogous statement is valid for k, being the Hamiltonian. However, if we choose k, to be the Hamiltonian, then the Casimir element q would be the fourthorder curve in the variables k, and kZ. The fourth-order curves are known to be parametrized in terms of elliptical functions, so the QAW(3) algebra also provides the theoretical foundation for systems which are integrable in terms of elliptical functions!
6
GRANOVSK~I,~UTZENKO, 3. QUANTUM
AND ZHEDANOV
VERSION-ALGEBRA
QAW(3)
In the preceding section we have constructed the classical algebra QAW(3) with the generators being the classical dynamical variables. Let us find the quantum analog of this algebra, replacing the classical variables ki by the operators K,. One cannot, however, merely transfer the relations (2.12) to the quantum case because of noncommutativity of the generators Ki. We need, therefore, the rule of ordering for these generators. One can treat the generators K, and Kz to be the Hermitian operators, then KS is anti-Hermitian, and the commutators [k;, rC,] and [K,, K,] are both Hermitian. If one is restricted to the real values of the structure constants then the right-hand sides’ of these commutators should contain only Hermitian terms. Thus, the transition from the classical to the quantum case can be performed, for example, by k,k; + K2K,K2; 2k,k,+
k;kz -+ K,K,K, (3.1)
(K,, K,),
where ( ., . > denotes the anticommutator. So we have the following “quantum”
(or commutator)
version of algebra
QAW(3) C43: CK,, KJ=& [ru,, KJ=2RK,K,Kz+A,(Kl,
K,) +A,K;+C,K,+DK,+G1
(3.2)
CK3,K1]=2RK1KZK1+A1K:.+A2{K1,KZ}+C2K2+DK1+G2
It is not diflicult to verify that the Jacobi identity
I[K,‘CKz,K,ll+
C&C&, &II + CK,CK,, JGII =O
(3.3)
is valid for Eq. (3.2). One can also find the Casimir operator Q commuting with all the generators Ki (replacing the classical terms by quantum ones using the rule (3.1)). However, the corresponding coefficients do not wholly coincide with the classical ones: Q=R(K,K;K,+K,K;K,)+(2-R)(A,K,K,K,+A,K,K,K,) +K$+(l-R)(C,Kf+CzK;)+(D-A,A,){K,,K,} +(G,(2-R)--*C,)K,+(G,(2-R)-C,A,)K,.
The obtained algebra QAW(3) possesses some important to construct the ladder representation.
(3.4)
properties allowing one
MUTUAL
7
INTEGRABILITY
Let $, be the eigenstate of generator K, with eigenvalue A,: K&=44,.
(3.5)
Then one can construct the new eigenstate GPSin the form
(3.6)
II/,,=(K*r(Kl)+K3vl(Kl)+i(K,))~p,
where t(K,), q(K,), [(K,) are some functions to be established. For the new eigenvalue A,,, one has the relation, which follows from (3.2) and
(3.6), (A,-&)'+2R~,~,~
+A,(I,+1,,)+
C2=O.
(3.7)
If the I, is fixed then the quadratic equation (3.7) yields two different values for A: II,. and A,,“. One can define p’ = p - 1, p” = p + 1. With this parametrization operator K, is three-diagonal and operator K, is two-diagonal:
The explicit expression for the spectrum 1, and the matrix coefficients up, 8, of the representation (3.8) can be obtained directly from the commutation relations (3.2) of the QAW(3) (for details of calculations see [4]). One can note that the spectrum I, is an exponential (also trigonometric or hyperbolic) function of p if R # 0, a quadratic function of p if R = 0 (but A, # 0), and a linear function of p if R=A,=O (but C,
Now we are ready to give the explanation of the denotation QAW(3). Let us take the representation of QAW(3) with discrete spectrum of I, (p = pl, p, + 1, ...) and P.~ being the initial value which satisfies the condition up, = 0. Then the overlap functions of two bases (S 1 p ) = (4, 1 II/,) are expressed in terms of (so-called) Askey-Wilson polynomials P,( pL,) of argument pS [6]:
C-JI P> = (s I Pl> (for details see [4]).
P,(PL,)Y
n=O, 1,2,...
(3.11)
8
~RA~OVSKII,~UTZENKO,
AND ZHEDANOV
If R # 0 then we have the Askey-Wilson q-polynomials expressing via q-hypergeometric function 4@3(q) [S]. In the case of Racah algebra QR(3) we obtain the Racah-Wilson poiynomials which can be expressed in terms of hy~rgeomet~~al function 41;;(f) [7]. For the Hahn SH(3) and Jacobi QJ(3) algebras we have the Hahn 3Fz (1) and Jacobi *F,(x) polynomials. Finally, for the Lie algebra (R = A, = A, = 0) we have the remaining classical orthogonal polynomials-both of discrete and continuous arguments. So the QAW(3) algebra is the dynamical symmetry algebra for all the known orthogonal polynomials. This statement was first established in [4] (for the case of Lie algebras see [S, 9 3). Recently it was shown that the QR(3) algebra is the hidden symmet~ algebra for oscillator and Kepler potentials in the space of constant curvature [3]; the Hahn algebra QH(3) serves as the hidden symmetry for Hartmann-like potentials [lo]; and the Jacobi algebra QJ(3) is the dynamical symmetry algebra for all the exactly solvable one-dimensional potentials of the Schroedinger equation [2]-this last case will be considered in the following sections. The correspondence between “quantum” QAW(3) and “classical” QAW(3) versions of the same algebra can be elucidated by the following prodecure. Let us denote by Vim” (i.e., classical limit) the transition from the quantum operator N to corresponding classical dynamical variable n, which takes place when h + 0: clim N = n.
(3.12)
Then for variables k, one has k, = clim K, ;
k, = clim( iK#).
k2 = clim Kz ;
The correspondence between commutators Dirac procedure:
(3.13).
and Poisson brackets is given by the
(ki, kK) = clim(i[&,
(3.14)
K&B).
The correspondence between classical and quantum structure parameters is pc = ,timo(- P,/fi2)
where p,(P,) Casimir).
is any classical (quantum)
parameter
(3.15) (i.e., structure
4. THE REPRESENTATIONS OF JACOBI ALGEBRA
constant or
QJ(3)
In this section we examine the representations of the quadratic Jacobi algebra QJ(3) because it is the algebra that plays the important role in the theory of exactly solvable potentials [2, 31.
MUTUAL
INTEGRABILITY
9
We recall that Jacobi algebra is the special case of QAW(3) with R = A, = C, = 0 (see Table I). It is convenient to reduce the Jacobi algebra to the most simple (“canonical”) form by means of transformations (4.1)
where ui, vi are arbitrary real parameters. Applying the transformation (4.1) one can always achieve the case: C2 = D = 0, A, = -2. So let us rewrite the Jacobi algebra in the “canonical” form (compare (3.2)):
CK2, KJ] = -2K; [K3, K,l=
-2(K,,
+ G,
(4.2)
K,} +G,.
The Casimir operator of the algebra (4.2) has the form Q=K:-2{K,,K,}+4K;+2G,K,+2G2K2.
(4.3)
For A,,, tx,,, 8, (see (3.8)) we have the expressions [2]
& = P(P+ 1)
(4.4)
B, = G,/~P(P + 1) (4.5)
where pI and p2 are the roots of the quartic equation 8G, p4 - 4Qp’ + G; = 0.
(4.6)
Because 8G, pf pi = Gi the roots p1 and p2 may be real and complex, depending on the sign of parameter G,. There are three different cases for all possible variants of representations: First, if G, is positive, we obtain three representation series: p1 and p2 are both real, 0 < p, < p2 (a) discrete series D+ for which IpI > p2 (p =p2, p2 + 1, p2 + 2, ... or, equivalently, p = -p2, -p2 - 1, ...). In this infinite-dimensional representation the spectrum 1, is purely discrete; (b) mixed series M+ consisting of a discrete part -p, < p < -i and of a continuous part p = iq - 4 (q is an arbitrary real parameter), joining together at the point p = - 1. p, and p2 have the imaginary part
(c)
representation
C+ with continuous spectrum p = iq - 4.
IO
GRANOVSKII,
LUTZENKO,
AND
ZHEDANOV
Second, G, is negative. In this case one of the roots (pi) is real and the second (pz) is imaginary. Here we have only mixed series M-, containing the discrete spectrum --pr < p < - +j and the continuous one p = iq - 1. Finally, if Gi = 0 we have the degenerate case of QJ(3) which may be reduced to the Lie algebra (see Appendix 1). Only mixed series MO exists in this case (with the same properties as in previous points). (Note that there are no finite-dimensional unitary representations of Jacobi algebra.) There are some interesting events occurring when p + i/2 (or pi --+ p2) with more or less substantional metamorphosis of spectra and reps but we will not pursue these questions. For non-degenerate representations the overlap functions (s 1 p> (see (3.11)) could be expressed in terms of the Gauss hypergeometric function 2F, (see [2]). Indeed, consider an alternative representation of QJ(3) in some basis wk such that both operators Ki and K2 are two-diagonal: Kl ok = akmk - 1 + b@,$
(4.7a)
&%=dk+1%+,
(4.7b)
+
gkOk>
k = 0, 1, 2, ... .
One can easily verify that such a basis really exists. The matrix elements ok, b,, dk, g, depend on the type of representation of QJ(3). If G, = +2 one has akdk = 2k(k + 5) b,=(k+k,)(k+k,-tl)
(4.8)
g,=l
where < =p2- p,, k, = p2 for the D+ M+ series. If Gi = -2 then
series and < = p, -p2,
k, = -pl
for the
iZkdk = 2ik(k + g)) bk=(k-k,)(k-kl
+ 1)
(4.9)
gkzii,
where 4;=pI--pt;k,= -pl. In the formulae (4.8) and (4.9) one of the matrix elements ak, dk can be arbitrary. The simplest case is dk E 1. To find the overlap functions (x 1p), where x is the eigenvalue of K2 let us expand the eigenfunction I,&, on the basis w&x), cx
1 p>
=
tip(x)
=x
ck~k(x)v k
(4.10)
MUTUAL
where ok(x)
11
INTEGRABILITY
is chosen to fulfil the relation (4.8b). For the case G, = +2, Ok(X)’
(x- l)k
(4.11a)
and, for the case G, = -2, Ok(X)=(X-iy.
(4.11b)
Substitution of (4.10) into the (4.7a) yields the recurrent coefficients Ck :
equation
for the (4.12
Ck+l/Ck=(~p-bk)lak+l. Solution of (4.12) has the form Ck = Cd - WY (k, - P)k (P + k, + 1 Mk!(l+
1 )k,
(4.13
where (t),=t(t+l)...(t+k-1) is Pochhammer symbol and C, is an arbitrary constant. From (4.10) and (4.13) one immediately obtains the q,(x) expressed in terms of Gauss hypergeometrical function, @p(x) = co zF1(4 B; Y I Y(X))9
(4.14)
where the hypergeometrical parameters LY,/I, y and argument y(x) depend on the representation type of QJ(3). For the discrete series D+ these parameters have the form a=~,-p=
-n,
Y(X) = (1 - x)/Z
For the mixed series M+ CI= -p,-p=
1,
/?=n+2p,+
Y=P2-P1+6
n=O, 1,2, ... .
(4.15)
we have -n,
B=n-2p,+
1,
Y=P1-P2+L
(4.16a)
Y(X) = (1 -x)/2
for the discrete part of spectrum and ct= -pl+rq+;,
j?= -p+q+;, Y(X) = (1 - x)/2
for the continuous
part of spectrum.
Y=PI-P*+i
(4.16b)
12
GRANOVSKII,
LUTZENKO,
Finally, for the mixed series Ma= -p*-p=
-n,
AND
ZHEDANOV
one obtains the results, /?=n-2pr+
1,
Y=P,-ilP*l+L
(4.17a)
y(x) = (1 + ix)/2, if one deals with the discrete part of the spectrum, and B= -PI---iq+$,
a= -p1+iq+;,
~=~1-il~21+l~
y(x) = (1 + ix)/2
(4.17b)
if one deals with the continuous part of the spectrum. For the degenerating case (G, = 0) the Jacobi algebra can be transformed into the Lie algebra (see Appendix 1) and the corresponding functions $,(x) are expressed in terms of the degenerating hypergeometrical function I F, . We have not considered here the case of the C+ series, because there are some “pathological” spectral properties in the physical realizations of this series, so the representations of this series should be considered separately. Note, that the basis wK(x) can be multiplied by the arbitrary functionf(x). Our choice f(x) = 1 corresponds to the operator K, being the purely hypergeometrical second-order differential operator. We have seen that analysis of the representations of Jacobi algebra allows one to construct the overlap (wave) functions (x 1$) and to express them in terms of hypergeometric functions. One can say, that Jacobi algebra is the “dynamical symmetry” of the hypergeometric equation-this algebra provides purely algebraic definition of $,(x) without solving the differential equation! The name “Jacobi algebra” is justified by the fact that, in the case of the discrete spectrum, the corresponding polynomials coincide with Jacobi polynomials-the most general orthogonal polynomials to which the Gauss hypergeometric function can be reduced [ 111).
5. JACOBI
ALGEBRA
AND
THE
“MANNING
MYSTERY”
Now we are ready to explain the “Manning mystery” connected with the exactly solvable potentials in quantum and classical mechanics. To this aim let us consider the realization of generators, K, = F(x)(p*/2
K* = z(x),
+ U(x) - W) = F(x)(H-
W)
(5.la) (5.lb)
where p is either classical or quantum (i.e., p= --ia,) momentum, H is a one-dimensional Hamiltonian with U(x) being the potential; F(x), z(x) are some functions, W is the arbitrary real parameter (the units are such that m = h = 1).
MUTUAL
INTEGRABILITY
13
Our main result is the following: if the same potential U(x) simultaneously satisfies both classical and quantum Jacobi algebra in realization (5.1), then (1) in the classical picture the variable z(x) is the elementary functions of the time (see Section 2) z, connected with the “Hamiltonian” Z?= F(x)H, i.e., dzjdz = (ff, z) = F(x)(H, z).
(5.2)
It is clear that the “true” time t is connected with r by the relation dt/dz = F(x).
(5.3)
So, returning to the ordinary time is performed by the simple change of variable (5.3); (2) in the quantum picture the eigenfunctions can be expressed in terms of the Gauss hypergeometrical function *F, and the eigenvalues of H are determined by a two-term recurrent relation (see Section 4). Both (1) and (2) statements follow from the concept of mutual integrability (Section 2) and from the analysis of QJ(3) representations (Section 4). We see that these statements resolve the Manning problem-the Jacobi algebra QJ(3) is a single non-trivial construction underlying both the classical and the quantum integrability. One can note that this situation resembles that in the Kepler problem, where the same algebra O(4) describes both classical and quantum degeneration [ 121. The analysis of the representation (5.1) is a rather complicated problem. We introduce the physically justified assumption that functions z(x), F(x), U(x) do not depend on the “external” spectral parameter W. Then we arrive at the relations (for details of the calculations see Appendix 2) z”(X) = -2n;(z)/n,(z)
(5.4a)
4~) = 71&Y27c,(z)
(5.4b)
U(x) = -?t&-cj + w,
(5.4c)
where n,(z) = az* + g,;
7-c*= c*z* + 2g,z - q
n3 = drc2/dW.
(5.5)
The relations (5.4) coincide with those of the Natanzon papers [13], providing the reduction of the Schroedinger equation to the hypergeometrical one. This reduction is not surprising since the QJ(3) algebra leads to the hypergeometrical equation. That is the algebraic reason for the Natanzon scheme. However, the conditions (5.4) are very broad because they include some quantum potentials which do not coincide with the classical ones. So let us demand that the same potential U(x) obeys both classical Qj(3) and quantum QJ(3) Jacobi
14
GRANOVSKII,
LUTZENKO, TABLE
AND ZHEDANOV II
Representations and Potentials Reps D+,Mf
MMO
U(x)
F(x)
(Name)
Wx)) th(x) cth(x)
1 1 ch2(x) sh*(x)
Poeschl Poeschl Eckart Eckart
Mx) sh(x)
cos2(x) 1
Eckart Poeschl
4x)
s tg* x + c ctg* x sth*x+ccth*x sth’x+cthx s cth* x + c cth x S tg* x + c tg x (s sh x + c)/ch’ x sx*+cx-2
+cx-* s exp( -x) + c exp( -2x) SX-I
cos(2x))
X-2
X2
X-’
X2
1
w(x)
oscillator Coulomb Morse
Note. Here are enumerated only physically distinct potentials which do not reduce to one another by linear transformation of argument x (with real parameters). The names of the potentials differ from those generally accepted-we stress the two different classes: the “Poeschl” class with F(x) = 1 and the “Eckart” class with F(x) # const.
algebras. It is not difficult to see that this condition is valid only if the roots of n3(z) coincide with the roots of xi(z) (see Appendix 2). Then we obtain the nine classes of physically distinct potentials connected with different representation series of QJ(3). These classes of potentials together with corresponding representations are enumerated in the Table II. 6. JACOBI ALGEBRA
AS DYNAMICAL
SYMMETRY
In the previous section we have showed that the Manning observation about classical and quantum exact integrability is explained by means of quadratic Jacobi algebra-Qj(3) in classical and QJ(3) in quantum mechanics. One can slightly modify the meaning of “exact integrability” in classical mechanics in terms of the Hamilton-Jacobi equation. Indeed, let the classical dynamical variables
k, =F(x) ;+ U(x)- w ( > kz = z(x)
obey the classical Jacobi algebra Qj(3). Consider the Hamilton-Jacobi (dS/dx)* + U(x) = w. 2
equation (6.1)
It is easily seen that the action S(z) is an elementary function of z. So we return to the initial treatment of Manning [l].
MUTUAL
INTEGRABILITY
15
Now we can formulate the statement: if the potential U(X) obeys both classical and quantum Jacobi algebra then the classical action S(z) is an elementary function of z(x) and the quantum wave function G,(z) is expressed in terms of the Gauss hypergeometrical function ,F,(( 1 - z(x))/2). One should note that this statement is more restrictive than the initial Manning observation. Indeed, in Manning’s treatment the possibility of expressing the action S(z) in terms of elementary functions implies the possibility to reduce the Schroedinger equation to the hypergeometrical one. However, there exist potentials U(x) obeying the classical Jacobi algebra Qj(3) (and, consequently, allowing the solution in elementary functions) but not obeying the quantum Jacobi algebra QJ(3) (and, consequently, not allowing reduction to the hypergeometrical equation). These potentials are obtained if the roots of the polynomial rcj(z) do not coincide with the roots of n,(z) (see (5.4)). This “pathological’ class of potentials must be considered separately. Let us call all the potentials in Table II “Manning potentials.” Then one can say that Jacobi algebra is the dynamical symmetry algebra for “Manning potentials.” By “dynamical s~metry” we mean the case when the Schroedinger equation with potential U(x) is reduced to the eigenvalue problem for one of the algebra generators (say, K,). Note that numerous attempts had been made to treat the “Manning potentials” from the dynamical symmetry point of view ([ 14-181 and others). However, there are two defects in their constructions: (1) no attempts had been made to treat the dynamical symmetries of exactly solvable potentials from the classical point of view; (2) only Lie algebras have been used to construct the dynamical symmetry. But the Schroedinger equation for the “Manning potentials” cannot be reduced to the eigenvalue problem for the Lie algebra generators (excluding the degenerate cases: oscillator, Coulomb, and Morse potentials-see [ 143). Therefore authors [14-181 had reduced the corresponding Schroedinger equation to the eigenvalue problem for some Casimir operators in different Lie algebras, like SU( 1, l), S0(2,2), etc. [ 17, 181. In our opinion their approach is complicated and does not satisfy the definition of dynamical symmetry. In our approach all the “Manning potentials” arise in frames of one quadratic Jacobi algebra either in classical or in quantum mechanics. This Jacobi algebra can be succesfully used to find all the needed physical characteristics of potentials: energy spectrum, wave functions, and scattering matrix-see [2]-dealing only algebraically, without integrating the differential equations. 7.
CONCLUSION
We have shown that the concept of mutual integrability is very useful in the analysis of systems with dynamical symmetries. We have demonstrate this concept 595/217/l-2
16
GRANOVSKII,
LUTZENKO,
AND
ZHEDANOV
applying it to the exactly solvable “Manning potentials.” The “Manning mystery” is resolved using the quadratic Jacobi algebra which has the same structure in the classical and quantum mechanics. One can note that the most general algebra QAW(3), obeying the condition of mutual integrability, is the dynamical symmetry of all possible problems where Askey-Wilson polynomials (and their specializations) arise as eigenfunctions. Probably the concept of mutual integrability should have applications in the theory of so-called quantum algebras where the noted polynomials arise. In the following publications we plan to examine the representations and applications of Racah QR(3) and Hahn QH(3) algebras and to elucidate the intimate connections of QAW(3) with so-called quantum algebras.
APPENDIX 1
Now it is an obvious thing that Lie algebras are just a particular case of quadratic algebra-it is sufhcient only to annulate some senior structure constants in commutation relations (3.2). But there is also one non-trivial case pointed out in Section 4--the so-called degenerate case of Jacobi algebra QJ(3), where the right-hand side of commutator [K2, K3] is the complete quadrat, i.e., if D2-4A2G1
=O.
It is obvious that by means of shifts and dilatations of initial generators K, and K2, one can always approach the canonical commutation relations in the degenerate case,
CL &I = -=:, CK,,&I =&r CK,,K,I = -2{&, &) +G,
(Al.1)
with the Casimir operator being (Alla)
Q=K;--4KzKlKzf2G2Kz.
Let us show that the quadratic terms may be eliminated by means of transition to the new generators N, = {K,, K,),
N2= K,?
(A1.2)
The third generator is [N1,N,J=N,=
-(K,,K,‘~=
-2K,‘K,+2.
(A1.3)
Here we have used an identity [K3, K,‘]
= -2.
(A1.4)
MUTUAL
17
INTEGRABILITY
The same identity also gives (A1.5)
[N,, NJ = -4N,. The last commutator [Nj, N,] = [ -2K;‘K,,
K, + 2K*K,]
is reduced (again with the aid of (A1.4), (Al.la))
to
[N3, N,] = 4G, - 4QN, - 4N,. Thus
CN,, NJ =N,;
[NJ, N,]=4G,-4QN,-4N,;
[N,, NJ = -4N,
(A1.6)
and we have the linear right-hand sides in all three commutators. The Casimir operator & of this Lie algebra has the form (A1.7)
Q=gikN,Nk+gG,N,.
The type of Lie algebra obtained depends on the structure of the Killing-Cartan tensor g, :
gik
.
=
(A1.8)
Taking into account that N, and N, are Hermitian operators but that N, is anti-Hermitian, we have that the algebra (A1.6) is isomorphic to the Lie algebra w, 1).
2
APPENDIX
Consider the classical problem. Let k, = I;,(x)(p’/2
+ u,. - W)
(A2.la)
k, = z(x) (A2.lb) k, = F,(x) z’(x) P, where the subscript “c” denotes the classical picture.
18
GRANOVSKII,
LUTZENKO,
AND
ZHEDANOV
The ki are supposed to satisfy the relations of the classical Jacobi algebra Qj(3), W,,k*)=k3 (A2.2)
(h, kJ = ak: + g, (k3, k,) = 2aklk2 + c2k2 + g2
(the structure Constance d, c, can be chosen to equal zero-see Section 4) with the Casimir element q=k:+2ak,k:+c2k:+2g,k,
Substitution
+2g,k,.
(A2.3)
of (A2.1) into (A2.2) and (A2.3) yields F,(x) L2(x) = -?T1(z)
(A2.4a)
2I;,(x)(U,(x) - w n,(z) = -n,(z),
(A2.4b)
where x1(z) = az2 + g,
(A2Sa)
x2(z) = c*z2 + 2g,z - q.
(A2Sb)
There are two relations (A2.4) for the three functions z(x), F(x), and U(x). So at least one of the functions, say z(x), can be arbitrary and we need one more requirement; namely, all the noted functions are independent of parameter W. Differentiating the relations (A2.4) on W one obtains
a71,law=o
(A2.6a)
an2/aw= X,(Z),
(A2.6b)
where 7Q(z) = uz2 + pz + y is an arbitrary second-order polynomial From (A2.4) and (A2.6) one obtains
u,-
with the coefficients independent of W.
ZI2= -2?+(z)/n3(z)
(A2.7)
w=
(A2.8)
-?r2(z)/7c3(z)
Formulae (A2.7) and (A2.8) yield the solution of the classical problem. Consider now the quantum problem. Let K,, K2, K3 be the following operators: K, = F&x)( -ii2 6Z/2 + u, - W) K2 = z(x)
K, = -t?&(X)(Z'(X) a, + 2’9).
(A2.9)
MUTUAL
19
INTEGRABILITY
These operators are supposed to satisfy the commutation algebra QJ( 3),
relations of Jacobi
CK,, &I = K, (A2.10)
[K,, KJ = AK: + G,
CK,,K,l=A{K,,K?)+CzK,+G, with the Casimir operator (A2.11)
Q=A{K,,K;}+K~+(A2+C~)K:+2G1K,+2G2K2.
According to (3.15) one can parametrize classical ones: c,=
A = --ah2;
Substitution
-c2h2;
G1.2
the quantum = -g,.,fi’;
parameters Q= -qh2.
via the (A2.12)
of (A2.9) into (A2.10) and (A2.11) yields two relations, F&x) z’2(x) = -7c1(z) 2qxw,
-
w n1(z) =
-n2(z)
(A2.13a) + &),
(A2.13b)
where the polynomials rc,(z) and rc2(z) coincide with the classical ones and d(z) is the specific additional quantum term: d(z) = h2[F22”’ + 2l;z’(Fz”)’ + 4aF(z’2 + zz”) + 4a222]/4.
(A2.14)
We see that F,(x) = F,(x) = F(x), but, in general, U,(x) # U,(x). From (A2.13b) one can conclude that the requirement of coinciding U,(x) with U,.(x) is reduced to the obvious “form-invariance” condition 0)
= %(Z),
(A2.15)
where ~~(2) is an arbitrary second-order polynomial. The condition (A2.15) can be rewritten in the compact form u’(z) + u2(2)/4 = ?Tn,(z)/7t:(z),
(A2.16)
where u(z) = i ln[zf2(x)]. The conditions of W-independence in the quantum case coincide with the classical ones (A2.6). It is clear that condition (A2.16) is compatible with (A2.7) if and only if the polynomial rc3(z) does not have roots different from those of the polynomial xi(z). Looking over all the possible variants we arrive at Table II.
20
GRANOVSKII,
LUTZENKO,
AND
ZHEDANOV
Note added in proof: As the referee has pointed out, it was S. Lie who first introduced and exploited general algebras with non-linear commutation relations (Math. Ann. 8 (1874/1875), 214). The authors are grateful to the referee for drawing their attention to this reference.
REFERENCES 1. M. MANNING, Pkys. Rev. 48 (1935), 161. 2. YA. I. GRANOVSKII, A. S. ZHEDANOV, AND I. M. LUTZENKO, Sov. Phys. JETP 72 (1991), 205. 3. YA. I. GRANOVSKII, A. S. ZHEDANOV, AND I. M. LUTZENKO, Teor. Mat. Fiz., to be published. [Russian] 4. YA. GRANOVSKII AND A. S. ZHEDANOV, Preprint DonFTI-89-7, 1989. 5. E. K. SKLYANIN, Funcr. Anal. Appl, 16 (1982), 266; 17 (1983), 273. 6. R. ASKEY AND J. WILSON, Mem. Am. Math. Sot. 54 (1985), 1. 7. J. WILSON, SIAM J. Math. Anal. 11 (1980), 690. 8. YA. I. GRANOVSK~ AND A. S. ZHEDANOV, Izv. VUZOV. Fiz. No. 5 (1986), 60. [Russian J 9. PH. FEINSILVER, Acta Appl. Math. 13 (1988), 291. 10. 0. F. GAL’BERT, YA. I. GRANOVSKII, AND A. S. ZHEDANOV, Phys. Left. A 153 (1991), 171; YA. I. GRANOVSKII, A. S. ZHEDANOV, AND I. M. LUTZENKO, J. Phys. A: Math. Gen. 24 (1991), 3887. Transcendental Functions,” Vol. 2, McGraw-Hill, New 11. H. BATEMAN AND A. ERDELYI, “Higher York, 1955. 12. J. P. ELLIOTT AND P. G. DAWBER, “Symmetry in Physics,” Vol. 2, Macmillan, London, 1979. 13. G. NATANZON, Vestn. Leningr. Univ. 10 (1971), 22; Tear. Mar. Fiz. 38 (1979), 146 (English transl.). 14. P. CORDERO, ET AL., Nuovo Cimento A 3 (1971), 807; B. G. KONOPEL’CHENKO AND Yu. B. RUMER, Dokl. Akad. Nauk SSSR 220 (1975), 58. [Russian] 15. G. G. GHIRARDI, Nuovo Cimento A 10 (1972), 97. 16. Y. ALHASSID, F. G~~RSEY, AND F. IACHELLO, Phys. Rev. Lett. 50 (1983), 873. 17. A. FRANK AND K. B. WOLF, J. Math. Phys. 26 (1985), 973. 18. J. Wu, Y. ALHASSID, AND F. G~RSEY, Ann. Phys. (N.Y.) 1% (1989), 163.
OF PHYSICS
217,
1-20
Mutual
YA.
I.
(1992)
Integrability, Quadratic Algebras, and Dynamical Symmetry
GRANOVSKII, Physics
I. M.
LUTZENKO,
A. S. ZHEDANOV
AND
Department, Donetsk University, Donetsk, 340055, Ukraine
Received September 11, 1991
The concept of mutually integrable dynamical variables is proposed. This concept leads to the quadratic Askey-Wilson algebra QAW(3) which is the dynamical symmetry algebra for all problems where the most general “classical” polynoials arise. In classical mechanics the algebra of the same structure describes the time evolution of dynamical variables in terms of elementary functions. We apply the special case of QAW(3kJacobi algebra-to describe the dynamical symmetry of exactly solvable potentials and to resolve the “Manning mystery”-the intimate relation between classical and quantum exactly solvable potentials. :D 1992 Academic Press, Inc.
1. INTRODUCTION More than a half-century ago M. Manning observed [ 1] that for each exactly solvable quantum problem there exists an integrable classical analogue, and, moreover, it is integrable in terms of elementary functions. Why does this happen? The obvious simplicity of this question should not hide the deep problems standing behind it. First, it is a problem of the homomorphism between quantum and classical approaches to the same object. The primary answer to this question is the quasiclassical treatment of the lines of the Bohr-Sommerfeld quantization rules. Manning’s observation is related to the situation when quasiclassics are not sufficient-but does that exhaust the subject? And what is the extended principle of quantization? The generally accepted and simple requirement of quadratic integrability of the wave function works only for simple potentials (with only one singular point) and collapses for more complicated ones where the problem of monodromy arises. The second problem is tightly connected with the widespread dogma that both potentials-classical and quantum-are identical and U(r) only becomes an operator in quantum-mechanical treatment! Is this true, indeed, or may it be due to this transfer that the specific quantum corrections come up? Manning’s correspondence puts definite restrictions on the possible structures of potentials. GQO3-4916192$9.00 Copyright d 1992 by Academic Press, Inc. All rights of reproductmn in any lorm reserved.
2
GRANOVSKII,LUTZENKO,
AND ZHEDANOV
Third, we would like to have a constructive arrangement, such as a Feynman path integral or something else, that would realise the needed homomorphism. Unfortunately, the technical difficulties (only Gaussian integrals or only soliton-like potentials) bring these attempts again to the quasiclassical route. Fourth, and finally, there should exist an abstract scheme that branches to either the quantum or the classical version along the way of its realization. The obvious candidate might be the dynamical symmetry group because of its intimate connection between exact solvability and intrinsic symmetry. To guess the group structure that would embrace all of the wide spectrum of Manning’s problems is a very non-trivial job with low chances for success. Nevertheless, we present an attempt moving just along the lines of a grouptheoretical approach. It contains a further development of our papers [2,3 3 on dynamical algebras containing all the exactly soluble problems of quantum mechanics. In Section 2 we propose a concept of mutual integrability that allows us to construct the quadratic algebra QAW(3), describing the wide class of “exactly solvable” classical problems (i.e., the time dependence of variables is expressed in terms of elementary functions). In Section 3 a quantum analog of this algebra-QAW(3 jis introduced. This algebra describes all the problems where the most general Askey-Wilson polynomials (and their degenerations) arise as eigenfunctions. In Section 4 we concentrate on the representations of Jacobi algebra QJ(3) (which is the special case of QAW(3)). The representations mentioned are used in Sections 5 and 6 to describe all the exactly solvable potentials and to resolve the “Manning mystery.”
2. MUTUAL
INTEGRABILITY-~LASSICAL
QAW(3)
We introduce the concept of mutual integrability proceeding from the classical mechanics. Let ki be the family of dynamical variables defined on the phase space with Poisson brackets denoted by (., .). The time dynamics is defined by the standard Poisson equation ki = (H, kJ,
(2-l)
where H is the Hamiltonian. Now we concentrate our attention on the two dynamical variables denoted by k, and k2 and let k, be the Hamiltonian: H= k,. Let us demand that the equation for k2 has the form -it,
where F,(k,) and t&(k,)
= F,(k,) kz + @,(k,),
are arbitrary
functions of k,.
(2.2)
MUTUAL
3
INTEGRABILITY
Because k, = const it is clear that the solution of (2.2) is an elementary function of time, if if if
cl exp(ot) + c2 exp( -ot) + c3 cl exp(iot) + c2 exp( -jot) + c3
k,(t) =
1 c,t2+c2t+c,
F,-cO
(2.3)
F,>O
(2.4)
F, = 0,
(2.5)
where o(k,) = dm, c,-arbitrary functions of k,. One can write Eq. (2.2) in algebraic form. To this aim let us define the third dynamical variable k, as the Poisson bracket of k, and k,: G.6)
k,=(k,,k,)
As k, is the Hamiltonian
then I;,=k, (2.7)
i;, = k, = (k,, k3)
and Eq. (2.2) can be rewritten in the form (k,, k,)=FAk,)
k,+ @,(k,).
(2.8)
Now we define the property of “relative integrability.” Namely, we say that dynamical variable k, is integrable relative to k, if the relations (2.6) and (2.8) are taking place. Because of the arbitrariness of the functions F2 and Q2 this definition is valid for the wide class of dynamical variables. However, one can reduce this class if the “mutual” integrability takes place. We say that variables k, and k, are “mutually integrable” if they are simultaneously integrable, one relative to the other. The property of mutual integrability can be written in the algebraic form: (k,, kz)=k, (2.9)
(k,, k,) = Fl(kJ k, + @l(kd (k,, k, I= F,(k, 1 k, + @z(k, ).
Despite the fact that property of mutual integrability seems to be very simple, it gives rise to many non-trivial consequences concerning the dynamical symmetry of classical (and quantum) problems. First, it strictly constrains the function Fi, @, due to relations (2.9). The relations (2.9) define a nonlinear Poisson algebra of generators ki. These generators should obey the Jacobi identity: (k,(k,, k3)) + Mk,,
k,)) + Mk,,
W = 0.
(2.10)
Inserting (2.9) into (2.10) one obtains that Fi and Di should be, in general, the
4
GRANOVSKII,
polynomials variables),
LUTZENKO,
AND
ZHEDANOV
of the second order (we treat ki and k, as algebraically independent F&k,) = Zrk; + Za,k, + cl FJk,)
= 2rkf + 2azk, + c2
(2.11)
Ql(k2) = a,k; + dkz + g, @P,(k,) = a&: + dk, + g,,
wherer, a,,,, clv2, d, g,,, are arbitrary parameters. We can now write the defining relations for the algebra (2.9) in explicit form [4],
(k,, k,)=h (k2, k,)=2rkgk,
+2a,k,k2+azk~+r,k,+dk,+gl
(k3, k,) = 2rk:k,
+ al kf + 2a2k, k2 + c,kz + dk, + g,.
(2.12)
The relations (2.12) define the bi-quadratic algebra of generators ki. By “quadratic algebra” we mean the object being introduced by Sklyanin [S J: the Poisson brackets of the generators are expressible in terms of quadratic (and linear) combinations of the generators. In the case of Eqs. (2.12), the right-hand sides of the relations are quadratic relative to both the kl and k, generators. We see that the concept of “mutual integrability” is the most natural way for introducing quadratic algebra into the dynamics. However, the arguments for introducing the first examples of quadratic algebras (Sklyanin algebras) were quite different (see [5]). Moreover, the Sklyanin’s algebras have more complicated structure and in general do not obey the property of mutual integrability. It will be shown that quadratic algebras defined by (2.12) are playing the important role in almost all exactly solvable problems of classical and quantum mechanics. Now we propose an appropriate terminology for the different cases of algebra (2.12). If r # 0 we call this algebra QAW(3)-quadratic Askey-Wilson algebra with three generators (the meaning of this and the following notations will be clear in the next sections). If r = 0 but a, az # 0 we obtain the so-called quadratic Racah algebra Qr(3). If r = a1 = 0 the corresponding algebra is called Hahn algebra Qh(3). If r = a1 = c, = 0 one obtains the quadratic Jacobi algebra Qj(3). Finally, if al = a, = 0 one obtains the ordinary Lie algebra (with three generators) isomorphic to either su(2), su(1, l), or Heisenberg-Weyl (oscillator) algebra (see Table I). One can obtain the Casismir element q for the algebra (2.12): (q,k,)=(q,k,)=(q,k,)=O. Indeed, from Eq. (2.2) we have $ [k: t F,(k,) k; + 2Qj2(kl) k,] = 0
(2.13)
MUTUAL
5
INTEGRABILITY
TABLE
I
Nomenclature of Quadratic Algebras
QAW(3) QR(3) QW)
QJ(3) Lie
*
* *
0
t,
*
0
0
0
0
Fl
* *
ii ;
*
or
k: + FAk, ) k: + 2@,(k,) k, = W, ), where ((k,) is some function of k,. In analogous way, by choosing k, to be the Hamiltonian, relation
k: + F,(k) k: +2@,(h) k, = v(k,)
(2.14)
one obtains the (2.15)
with some function q(k,). Combining (2.14) and (2.15) one obtains the expression for the Casimir element q independent of time in both cases:
+c,k;+c,k:+2dk,k,+2g,kl+2g,k,.
(2.16)
It is not difficult to verify that expression (2.16) indeed yields the Casimir element-one needs to directly calculate the Poisson brackets (q, kJ. It is interesting to note that contrary to Lie algebras where the Casimir element is quadratic in the generators, our Casimir element q is wholly of the fourth order but quadratic separately in k,, k,, k,. In particular, if k, is the Hamiltonian, then k, = const and q defines the quadratic curve in the variables k, and k,. Because all the quadratic curves can be parametrized via the elementary functions we obtain one more interpretation of the mutual integrability. Indeed, the time t is the parameter providing the description of k2 in terms of elementary functions. An analogous statement is valid for k, being the Hamiltonian. However, if we choose k, to be the Hamiltonian, then the Casimir element q would be the fourthorder curve in the variables k, and kZ. The fourth-order curves are known to be parametrized in terms of elliptical functions, so the QAW(3) algebra also provides the theoretical foundation for systems which are integrable in terms of elliptical functions!
6
GRANOVSK~I,~UTZENKO, 3. QUANTUM
AND ZHEDANOV
VERSION-ALGEBRA
QAW(3)
In the preceding section we have constructed the classical algebra QAW(3) with the generators being the classical dynamical variables. Let us find the quantum analog of this algebra, replacing the classical variables ki by the operators K,. One cannot, however, merely transfer the relations (2.12) to the quantum case because of noncommutativity of the generators Ki. We need, therefore, the rule of ordering for these generators. One can treat the generators K, and Kz to be the Hermitian operators, then KS is anti-Hermitian, and the commutators [k;, rC,] and [K,, K,] are both Hermitian. If one is restricted to the real values of the structure constants then the right-hand sides’ of these commutators should contain only Hermitian terms. Thus, the transition from the classical to the quantum case can be performed, for example, by k,k; + K2K,K2; 2k,k,+
k;kz -+ K,K,K, (3.1)
(K,, K,),
where ( ., . > denotes the anticommutator. So we have the following “quantum”
(or commutator)
version of algebra
QAW(3) C43: CK,, KJ=& [ru,, KJ=2RK,K,Kz+A,(Kl,
K,) +A,K;+C,K,+DK,+G1
(3.2)
CK3,K1]=2RK1KZK1+A1K:.+A2{K1,KZ}+C2K2+DK1+G2
It is not diflicult to verify that the Jacobi identity
I[K,‘CKz,K,ll+
C&C&, &II + CK,CK,, JGII =O
(3.3)
is valid for Eq. (3.2). One can also find the Casimir operator Q commuting with all the generators Ki (replacing the classical terms by quantum ones using the rule (3.1)). However, the corresponding coefficients do not wholly coincide with the classical ones: Q=R(K,K;K,+K,K;K,)+(2-R)(A,K,K,K,+A,K,K,K,) +K$+(l-R)(C,Kf+CzK;)+(D-A,A,){K,,K,} +(G,(2-R)--*C,)K,+(G,(2-R)-C,A,)K,.
The obtained algebra QAW(3) possesses some important to construct the ladder representation.
(3.4)
properties allowing one
MUTUAL
7
INTEGRABILITY
Let $, be the eigenstate of generator K, with eigenvalue A,: K&=44,.
(3.5)
Then one can construct the new eigenstate GPSin the form
(3.6)
II/,,=(K*r(Kl)+K3vl(Kl)+i(K,))~p,
where t(K,), q(K,), [(K,) are some functions to be established. For the new eigenvalue A,,, one has the relation, which follows from (3.2) and
(3.6), (A,-&)'+2R~,~,~
+A,(I,+1,,)+
C2=O.
(3.7)
If the I, is fixed then the quadratic equation (3.7) yields two different values for A: II,. and A,,“. One can define p’ = p - 1, p” = p + 1. With this parametrization operator K, is three-diagonal and operator K, is two-diagonal:
The explicit expression for the spectrum 1, and the matrix coefficients up, 8, of the representation (3.8) can be obtained directly from the commutation relations (3.2) of the QAW(3) (for details of calculations see [4]). One can note that the spectrum I, is an exponential (also trigonometric or hyperbolic) function of p if R # 0, a quadratic function of p if R = 0 (but A, # 0), and a linear function of p if R=A,=O (but C,
Now we are ready to give the explanation of the denotation QAW(3). Let us take the representation of QAW(3) with discrete spectrum of I, (p = pl, p, + 1, ...) and P.~ being the initial value which satisfies the condition up, = 0. Then the overlap functions of two bases (S 1 p ) = (4, 1 II/,) are expressed in terms of (so-called) Askey-Wilson polynomials P,( pL,) of argument pS [6]:
C-JI P> = (s I Pl> (for details see [4]).
P,(PL,)Y
n=O, 1,2,...
(3.11)
8
~RA~OVSKII,~UTZENKO,
AND ZHEDANOV
If R # 0 then we have the Askey-Wilson q-polynomials expressing via q-hypergeometric function 4@3(q) [S]. In the case of Racah algebra QR(3) we obtain the Racah-Wilson poiynomials which can be expressed in terms of hy~rgeomet~~al function 41;;(f) [7]. For the Hahn SH(3) and Jacobi QJ(3) algebras we have the Hahn 3Fz (1) and Jacobi *F,(x) polynomials. Finally, for the Lie algebra (R = A, = A, = 0) we have the remaining classical orthogonal polynomials-both of discrete and continuous arguments. So the QAW(3) algebra is the dynamical symmetry algebra for all the known orthogonal polynomials. This statement was first established in [4] (for the case of Lie algebras see [S, 9 3). Recently it was shown that the QR(3) algebra is the hidden symmet~ algebra for oscillator and Kepler potentials in the space of constant curvature [3]; the Hahn algebra QH(3) serves as the hidden symmetry for Hartmann-like potentials [lo]; and the Jacobi algebra QJ(3) is the dynamical symmetry algebra for all the exactly solvable one-dimensional potentials of the Schroedinger equation [2]-this last case will be considered in the following sections. The correspondence between “quantum” QAW(3) and “classical” QAW(3) versions of the same algebra can be elucidated by the following prodecure. Let us denote by Vim” (i.e., classical limit) the transition from the quantum operator N to corresponding classical dynamical variable n, which takes place when h + 0: clim N = n.
(3.12)
Then for variables k, one has k, = clim K, ;
k, = clim( iK#).
k2 = clim Kz ;
The correspondence between commutators Dirac procedure:
(3.13).
and Poisson brackets is given by the
(ki, kK) = clim(i[&,
(3.14)
K&B).
The correspondence between classical and quantum structure parameters is pc = ,timo(- P,/fi2)
where p,(P,) Casimir).
is any classical (quantum)
parameter
(3.15) (i.e., structure
4. THE REPRESENTATIONS OF JACOBI ALGEBRA
constant or
QJ(3)
In this section we examine the representations of the quadratic Jacobi algebra QJ(3) because it is the algebra that plays the important role in the theory of exactly solvable potentials [2, 31.
MUTUAL
INTEGRABILITY
9
We recall that Jacobi algebra is the special case of QAW(3) with R = A, = C, = 0 (see Table I). It is convenient to reduce the Jacobi algebra to the most simple (“canonical”) form by means of transformations (4.1)
where ui, vi are arbitrary real parameters. Applying the transformation (4.1) one can always achieve the case: C2 = D = 0, A, = -2. So let us rewrite the Jacobi algebra in the “canonical” form (compare (3.2)):
CK2, KJ] = -2K; [K3, K,l=
-2(K,,
+ G,
(4.2)
K,} +G,.
The Casimir operator of the algebra (4.2) has the form Q=K:-2{K,,K,}+4K;+2G,K,+2G2K2.
(4.3)
For A,,, tx,,, 8, (see (3.8)) we have the expressions [2]
& = P(P+ 1)
(4.4)
B, = G,/~P(P + 1) (4.5)
where pI and p2 are the roots of the quartic equation 8G, p4 - 4Qp’ + G; = 0.
(4.6)
Because 8G, pf pi = Gi the roots p1 and p2 may be real and complex, depending on the sign of parameter G,. There are three different cases for all possible variants of representations: First, if G, is positive, we obtain three representation series: p1 and p2 are both real, 0 < p, < p2 (a) discrete series D+ for which IpI > p2 (p =p2, p2 + 1, p2 + 2, ... or, equivalently, p = -p2, -p2 - 1, ...). In this infinite-dimensional representation the spectrum 1, is purely discrete; (b) mixed series M+ consisting of a discrete part -p, < p < -i and of a continuous part p = iq - 4 (q is an arbitrary real parameter), joining together at the point p = - 1. p, and p2 have the imaginary part
(c)
representation
C+ with continuous spectrum p = iq - 4.
IO
GRANOVSKII,
LUTZENKO,
AND
ZHEDANOV
Second, G, is negative. In this case one of the roots (pi) is real and the second (pz) is imaginary. Here we have only mixed series M-, containing the discrete spectrum --pr < p < - +j and the continuous one p = iq - 1. Finally, if Gi = 0 we have the degenerate case of QJ(3) which may be reduced to the Lie algebra (see Appendix 1). Only mixed series MO exists in this case (with the same properties as in previous points). (Note that there are no finite-dimensional unitary representations of Jacobi algebra.) There are some interesting events occurring when p + i/2 (or pi --+ p2) with more or less substantional metamorphosis of spectra and reps but we will not pursue these questions. For non-degenerate representations the overlap functions (s 1 p> (see (3.11)) could be expressed in terms of the Gauss hypergeometric function 2F, (see [2]). Indeed, consider an alternative representation of QJ(3) in some basis wk such that both operators Ki and K2 are two-diagonal: Kl ok = akmk - 1 + b@,$
(4.7a)
&%=dk+1%+,
(4.7b)
+
gkOk>
k = 0, 1, 2, ... .
One can easily verify that such a basis really exists. The matrix elements ok, b,, dk, g, depend on the type of representation of QJ(3). If G, = +2 one has akdk = 2k(k + 5) b,=(k+k,)(k+k,-tl)
(4.8)
g,=l
where < =p2- p,, k, = p2 for the D+ M+ series. If Gi = -2 then
series and < = p, -p2,
k, = -pl
for the
iZkdk = 2ik(k + g)) bk=(k-k,)(k-kl
+ 1)
(4.9)
gkzii,
where 4;=pI--pt;k,= -pl. In the formulae (4.8) and (4.9) one of the matrix elements ak, dk can be arbitrary. The simplest case is dk E 1. To find the overlap functions (x 1p), where x is the eigenvalue of K2 let us expand the eigenfunction I,&, on the basis w&x), cx
1 p>
=
tip(x)
=x
ck~k(x)v k
(4.10)
MUTUAL
where ok(x)
11
INTEGRABILITY
is chosen to fulfil the relation (4.8b). For the case G, = +2, Ok(X)’
(x- l)k
(4.11a)
and, for the case G, = -2, Ok(X)=(X-iy.
(4.11b)
Substitution of (4.10) into the (4.7a) yields the recurrent coefficients Ck :
equation
for the (4.12
Ck+l/Ck=(~p-bk)lak+l. Solution of (4.12) has the form Ck = Cd - WY (k, - P)k (P + k, + 1 Mk!(l+
1 )k,
(4.13
where (t),=t(t+l)...(t+k-1) is Pochhammer symbol and C, is an arbitrary constant. From (4.10) and (4.13) one immediately obtains the q,(x) expressed in terms of Gauss hypergeometrical function, @p(x) = co zF1(4 B; Y I Y(X))9
(4.14)
where the hypergeometrical parameters LY,/I, y and argument y(x) depend on the representation type of QJ(3). For the discrete series D+ these parameters have the form a=~,-p=
-n,
Y(X) = (1 - x)/Z
For the mixed series M+ CI= -p,-p=
1,
/?=n+2p,+
Y=P2-P1+6
n=O, 1,2, ... .
(4.15)
we have -n,
B=n-2p,+
1,
Y=P1-P2+L
(4.16a)
Y(X) = (1 -x)/2
for the discrete part of spectrum and ct= -pl+rq+;,
j?= -p+q+;, Y(X) = (1 - x)/2
for the continuous
part of spectrum.
Y=PI-P*+i
(4.16b)
12
GRANOVSKII,
LUTZENKO,
Finally, for the mixed series Ma= -p*-p=
-n,
AND
ZHEDANOV
one obtains the results, /?=n-2pr+
1,
Y=P,-ilP*l+L
(4.17a)
y(x) = (1 + ix)/2, if one deals with the discrete part of the spectrum, and B= -PI---iq+$,
a= -p1+iq+;,
~=~1-il~21+l~
y(x) = (1 + ix)/2
(4.17b)
if one deals with the continuous part of the spectrum. For the degenerating case (G, = 0) the Jacobi algebra can be transformed into the Lie algebra (see Appendix 1) and the corresponding functions $,(x) are expressed in terms of the degenerating hypergeometrical function I F, . We have not considered here the case of the C+ series, because there are some “pathological” spectral properties in the physical realizations of this series, so the representations of this series should be considered separately. Note, that the basis wK(x) can be multiplied by the arbitrary functionf(x). Our choice f(x) = 1 corresponds to the operator K, being the purely hypergeometrical second-order differential operator. We have seen that analysis of the representations of Jacobi algebra allows one to construct the overlap (wave) functions (x 1$) and to express them in terms of hypergeometric functions. One can say, that Jacobi algebra is the “dynamical symmetry” of the hypergeometric equation-this algebra provides purely algebraic definition of $,(x) without solving the differential equation! The name “Jacobi algebra” is justified by the fact that, in the case of the discrete spectrum, the corresponding polynomials coincide with Jacobi polynomials-the most general orthogonal polynomials to which the Gauss hypergeometric function can be reduced [ 111).
5. JACOBI
ALGEBRA
AND
THE
“MANNING
MYSTERY”
Now we are ready to explain the “Manning mystery” connected with the exactly solvable potentials in quantum and classical mechanics. To this aim let us consider the realization of generators, K, = F(x)(p*/2
K* = z(x),
+ U(x) - W) = F(x)(H-
W)
(5.la) (5.lb)
where p is either classical or quantum (i.e., p= --ia,) momentum, H is a one-dimensional Hamiltonian with U(x) being the potential; F(x), z(x) are some functions, W is the arbitrary real parameter (the units are such that m = h = 1).
MUTUAL
INTEGRABILITY
13
Our main result is the following: if the same potential U(x) simultaneously satisfies both classical and quantum Jacobi algebra in realization (5.1), then (1) in the classical picture the variable z(x) is the elementary functions of the time (see Section 2) z, connected with the “Hamiltonian” Z?= F(x)H, i.e., dzjdz = (ff, z) = F(x)(H, z).
(5.2)
It is clear that the “true” time t is connected with r by the relation dt/dz = F(x).
(5.3)
So, returning to the ordinary time is performed by the simple change of variable (5.3); (2) in the quantum picture the eigenfunctions can be expressed in terms of the Gauss hypergeometrical function *F, and the eigenvalues of H are determined by a two-term recurrent relation (see Section 4). Both (1) and (2) statements follow from the concept of mutual integrability (Section 2) and from the analysis of QJ(3) representations (Section 4). We see that these statements resolve the Manning problem-the Jacobi algebra QJ(3) is a single non-trivial construction underlying both the classical and the quantum integrability. One can note that this situation resembles that in the Kepler problem, where the same algebra O(4) describes both classical and quantum degeneration [ 121. The analysis of the representation (5.1) is a rather complicated problem. We introduce the physically justified assumption that functions z(x), F(x), U(x) do not depend on the “external” spectral parameter W. Then we arrive at the relations (for details of the calculations see Appendix 2) z”(X) = -2n;(z)/n,(z)
(5.4a)
4~) = 71&Y27c,(z)
(5.4b)
U(x) = -?t&-cj + w,
(5.4c)
where n,(z) = az* + g,;
7-c*= c*z* + 2g,z - q
n3 = drc2/dW.
(5.5)
The relations (5.4) coincide with those of the Natanzon papers [13], providing the reduction of the Schroedinger equation to the hypergeometrical one. This reduction is not surprising since the QJ(3) algebra leads to the hypergeometrical equation. That is the algebraic reason for the Natanzon scheme. However, the conditions (5.4) are very broad because they include some quantum potentials which do not coincide with the classical ones. So let us demand that the same potential U(x) obeys both classical Qj(3) and quantum QJ(3) Jacobi
14
GRANOVSKII,
LUTZENKO, TABLE
AND ZHEDANOV II
Representations and Potentials Reps D+,Mf
MMO
U(x)
F(x)
(Name)
Wx)) th(x) cth(x)
1 1 ch2(x) sh*(x)
Poeschl Poeschl Eckart Eckart
Mx) sh(x)
cos2(x) 1
Eckart Poeschl
4x)
s tg* x + c ctg* x sth*x+ccth*x sth’x+cthx s cth* x + c cth x S tg* x + c tg x (s sh x + c)/ch’ x sx*+cx-2
+cx-* s exp( -x) + c exp( -2x) SX-I
cos(2x))
X-2
X2
X-’
X2
1
w(x)
oscillator Coulomb Morse
Note. Here are enumerated only physically distinct potentials which do not reduce to one another by linear transformation of argument x (with real parameters). The names of the potentials differ from those generally accepted-we stress the two different classes: the “Poeschl” class with F(x) = 1 and the “Eckart” class with F(x) # const.
algebras. It is not difficult to see that this condition is valid only if the roots of n3(z) coincide with the roots of xi(z) (see Appendix 2). Then we obtain the nine classes of physically distinct potentials connected with different representation series of QJ(3). These classes of potentials together with corresponding representations are enumerated in the Table II. 6. JACOBI ALGEBRA
AS DYNAMICAL
SYMMETRY
In the previous section we have showed that the Manning observation about classical and quantum exact integrability is explained by means of quadratic Jacobi algebra-Qj(3) in classical and QJ(3) in quantum mechanics. One can slightly modify the meaning of “exact integrability” in classical mechanics in terms of the Hamilton-Jacobi equation. Indeed, let the classical dynamical variables
k, =F(x) ;+ U(x)- w ( > kz = z(x)
obey the classical Jacobi algebra Qj(3). Consider the Hamilton-Jacobi (dS/dx)* + U(x) = w. 2
equation (6.1)
It is easily seen that the action S(z) is an elementary function of z. So we return to the initial treatment of Manning [l].
MUTUAL
INTEGRABILITY
15
Now we can formulate the statement: if the potential U(X) obeys both classical and quantum Jacobi algebra then the classical action S(z) is an elementary function of z(x) and the quantum wave function G,(z) is expressed in terms of the Gauss hypergeometrical function ,F,(( 1 - z(x))/2). One should note that this statement is more restrictive than the initial Manning observation. Indeed, in Manning’s treatment the possibility of expressing the action S(z) in terms of elementary functions implies the possibility to reduce the Schroedinger equation to the hypergeometrical one. However, there exist potentials U(x) obeying the classical Jacobi algebra Qj(3) (and, consequently, allowing the solution in elementary functions) but not obeying the quantum Jacobi algebra QJ(3) (and, consequently, not allowing reduction to the hypergeometrical equation). These potentials are obtained if the roots of the polynomial rcj(z) do not coincide with the roots of n,(z) (see (5.4)). This “pathological’ class of potentials must be considered separately. Let us call all the potentials in Table II “Manning potentials.” Then one can say that Jacobi algebra is the dynamical symmetry algebra for “Manning potentials.” By “dynamical s~metry” we mean the case when the Schroedinger equation with potential U(x) is reduced to the eigenvalue problem for one of the algebra generators (say, K,). Note that numerous attempts had been made to treat the “Manning potentials” from the dynamical symmetry point of view ([ 14-181 and others). However, there are two defects in their constructions: (1) no attempts had been made to treat the dynamical symmetries of exactly solvable potentials from the classical point of view; (2) only Lie algebras have been used to construct the dynamical symmetry. But the Schroedinger equation for the “Manning potentials” cannot be reduced to the eigenvalue problem for the Lie algebra generators (excluding the degenerate cases: oscillator, Coulomb, and Morse potentials-see [ 143). Therefore authors [14-181 had reduced the corresponding Schroedinger equation to the eigenvalue problem for some Casimir operators in different Lie algebras, like SU( 1, l), S0(2,2), etc. [ 17, 181. In our opinion their approach is complicated and does not satisfy the definition of dynamical symmetry. In our approach all the “Manning potentials” arise in frames of one quadratic Jacobi algebra either in classical or in quantum mechanics. This Jacobi algebra can be succesfully used to find all the needed physical characteristics of potentials: energy spectrum, wave functions, and scattering matrix-see [2]-dealing only algebraically, without integrating the differential equations. 7.
CONCLUSION
We have shown that the concept of mutual integrability is very useful in the analysis of systems with dynamical symmetries. We have demonstrate this concept 595/217/l-2
16
GRANOVSKII,
LUTZENKO,
AND
ZHEDANOV
applying it to the exactly solvable “Manning potentials.” The “Manning mystery” is resolved using the quadratic Jacobi algebra which has the same structure in the classical and quantum mechanics. One can note that the most general algebra QAW(3), obeying the condition of mutual integrability, is the dynamical symmetry of all possible problems where Askey-Wilson polynomials (and their specializations) arise as eigenfunctions. Probably the concept of mutual integrability should have applications in the theory of so-called quantum algebras where the noted polynomials arise. In the following publications we plan to examine the representations and applications of Racah QR(3) and Hahn QH(3) algebras and to elucidate the intimate connections of QAW(3) with so-called quantum algebras.
APPENDIX 1
Now it is an obvious thing that Lie algebras are just a particular case of quadratic algebra-it is sufhcient only to annulate some senior structure constants in commutation relations (3.2). But there is also one non-trivial case pointed out in Section 4--the so-called degenerate case of Jacobi algebra QJ(3), where the right-hand side of commutator [K2, K3] is the complete quadrat, i.e., if D2-4A2G1
=O.
It is obvious that by means of shifts and dilatations of initial generators K, and K2, one can always approach the canonical commutation relations in the degenerate case,
CL &I = -=:, CK,,&I =&r CK,,K,I = -2{&, &) +G,
(Al.1)
with the Casimir operator being (Alla)
Q=K;--4KzKlKzf2G2Kz.
Let us show that the quadratic terms may be eliminated by means of transition to the new generators N, = {K,, K,),
N2= K,?
(A1.2)
The third generator is [N1,N,J=N,=
-(K,,K,‘~=
-2K,‘K,+2.
(A1.3)
Here we have used an identity [K3, K,‘]
= -2.
(A1.4)
MUTUAL
17
INTEGRABILITY
The same identity also gives (A1.5)
[N,, NJ = -4N,. The last commutator [Nj, N,] = [ -2K;‘K,,
K, + 2K*K,]
is reduced (again with the aid of (A1.4), (Al.la))
to
[N3, N,] = 4G, - 4QN, - 4N,. Thus
CN,, NJ =N,;
[NJ, N,]=4G,-4QN,-4N,;
[N,, NJ = -4N,
(A1.6)
and we have the linear right-hand sides in all three commutators. The Casimir operator & of this Lie algebra has the form (A1.7)
Q=gikN,Nk+gG,N,.
The type of Lie algebra obtained depends on the structure of the Killing-Cartan tensor g, :
gik
.
=
(A1.8)
Taking into account that N, and N, are Hermitian operators but that N, is anti-Hermitian, we have that the algebra (A1.6) is isomorphic to the Lie algebra w, 1).
2
APPENDIX
Consider the classical problem. Let k, = I;,(x)(p’/2
+ u,. - W)
(A2.la)
k, = z(x) (A2.lb) k, = F,(x) z’(x) P, where the subscript “c” denotes the classical picture.
18
GRANOVSKII,
LUTZENKO,
AND
ZHEDANOV
The ki are supposed to satisfy the relations of the classical Jacobi algebra Qj(3), W,,k*)=k3 (A2.2)
(h, kJ = ak: + g, (k3, k,) = 2aklk2 + c2k2 + g2
(the structure Constance d, c, can be chosen to equal zero-see Section 4) with the Casimir element q=k:+2ak,k:+c2k:+2g,k,
Substitution
+2g,k,.
(A2.3)
of (A2.1) into (A2.2) and (A2.3) yields F,(x) L2(x) = -?T1(z)
(A2.4a)
2I;,(x)(U,(x) - w n,(z) = -n,(z),
(A2.4b)
where x1(z) = az2 + g,
(A2Sa)
x2(z) = c*z2 + 2g,z - q.
(A2Sb)
There are two relations (A2.4) for the three functions z(x), F(x), and U(x). So at least one of the functions, say z(x), can be arbitrary and we need one more requirement; namely, all the noted functions are independent of parameter W. Differentiating the relations (A2.4) on W one obtains
a71,law=o
(A2.6a)
an2/aw= X,(Z),
(A2.6b)
where 7Q(z) = uz2 + pz + y is an arbitrary second-order polynomial From (A2.4) and (A2.6) one obtains
u,-
with the coefficients independent of W.
ZI2= -2?+(z)/n3(z)
(A2.7)
w=
(A2.8)
-?r2(z)/7c3(z)
Formulae (A2.7) and (A2.8) yield the solution of the classical problem. Consider now the quantum problem. Let K,, K2, K3 be the following operators: K, = F&x)( -ii2 6Z/2 + u, - W) K2 = z(x)
K, = -t?&(X)(Z'(X) a, + 2’9).
(A2.9)
MUTUAL
19
INTEGRABILITY
These operators are supposed to satisfy the commutation algebra QJ( 3),
relations of Jacobi
CK,, &I = K, (A2.10)
[K,, KJ = AK: + G,
CK,,K,l=A{K,,K?)+CzK,+G, with the Casimir operator (A2.11)
Q=A{K,,K;}+K~+(A2+C~)K:+2G1K,+2G2K2.
According to (3.15) one can parametrize classical ones: c,=
A = --ah2;
Substitution
-c2h2;
G1.2
the quantum = -g,.,fi’;
parameters Q= -qh2.
via the (A2.12)
of (A2.9) into (A2.10) and (A2.11) yields two relations, F&x) z’2(x) = -7c1(z) 2qxw,
-
w n1(z) =
-n2(z)
(A2.13a) + &),
(A2.13b)
where the polynomials rc,(z) and rc2(z) coincide with the classical ones and d(z) is the specific additional quantum term: d(z) = h2[F22”’ + 2l;z’(Fz”)’ + 4aF(z’2 + zz”) + 4a222]/4.
(A2.14)
We see that F,(x) = F,(x) = F(x), but, in general, U,(x) # U,(x). From (A2.13b) one can conclude that the requirement of coinciding U,(x) with U,.(x) is reduced to the obvious “form-invariance” condition 0)
= %(Z),
(A2.15)
where ~~(2) is an arbitrary second-order polynomial. The condition (A2.15) can be rewritten in the compact form u’(z) + u2(2)/4 = ?Tn,(z)/7t:(z),
(A2.16)
where u(z) = i ln[zf2(x)]. The conditions of W-independence in the quantum case coincide with the classical ones (A2.6). It is clear that condition (A2.16) is compatible with (A2.7) if and only if the polynomial rc3(z) does not have roots different from those of the polynomial xi(z). Looking over all the possible variants we arrive at Table II.
20
GRANOVSKII,
LUTZENKO,
AND
ZHEDANOV
Note added in proof: As the referee has pointed out, it was S. Lie who first introduced and exploited general algebras with non-linear commutation relations (Math. Ann. 8 (1874/1875), 214). The authors are grateful to the referee for drawing their attention to this reference.
REFERENCES 1. M. MANNING, Pkys. Rev. 48 (1935), 161. 2. YA. I. GRANOVSKII, A. S. ZHEDANOV, AND I. M. LUTZENKO, Sov. Phys. JETP 72 (1991), 205. 3. YA. I. GRANOVSKII, A. S. ZHEDANOV, AND I. M. LUTZENKO, Teor. Mat. Fiz., to be published. [Russian] 4. YA. GRANOVSKII AND A. S. ZHEDANOV, Preprint DonFTI-89-7, 1989. 5. E. K. SKLYANIN, Funcr. Anal. Appl, 16 (1982), 266; 17 (1983), 273. 6. R. ASKEY AND J. WILSON, Mem. Am. Math. Sot. 54 (1985), 1. 7. J. WILSON, SIAM J. Math. Anal. 11 (1980), 690. 8. YA. I. GRANOVSK~ AND A. S. ZHEDANOV, Izv. VUZOV. Fiz. No. 5 (1986), 60. [Russian J 9. PH. FEINSILVER, Acta Appl. Math. 13 (1988), 291. 10. 0. F. GAL’BERT, YA. I. GRANOVSKII, AND A. S. ZHEDANOV, Phys. Left. A 153 (1991), 171; YA. I. GRANOVSKII, A. S. ZHEDANOV, AND I. M. LUTZENKO, J. Phys. A: Math. Gen. 24 (1991), 3887. Transcendental Functions,” Vol. 2, McGraw-Hill, New 11. H. BATEMAN AND A. ERDELYI, “Higher York, 1955. 12. J. P. ELLIOTT AND P. G. DAWBER, “Symmetry in Physics,” Vol. 2, Macmillan, London, 1979. 13. G. NATANZON, Vestn. Leningr. Univ. 10 (1971), 22; Tear. Mar. Fiz. 38 (1979), 146 (English transl.). 14. P. CORDERO, ET AL., Nuovo Cimento A 3 (1971), 807; B. G. KONOPEL’CHENKO AND Yu. B. RUMER, Dokl. Akad. Nauk SSSR 220 (1975), 58. [Russian] 15. G. G. GHIRARDI, Nuovo Cimento A 10 (1972), 97. 16. Y. ALHASSID, F. G~~RSEY, AND F. IACHELLO, Phys. Rev. Lett. 50 (1983), 873. 17. A. FRANK AND K. B. WOLF, J. Math. Phys. 26 (1985), 973. 18. J. Wu, Y. ALHASSID, AND F. G~RSEY, Ann. Phys. (N.Y.) 1% (1989), 163.