Letters in Mathematical Physics 33: 61-74, 1995. © 1995 Kluwer Academic Publishers. Printed in the Netherlands.
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Harmonics on Hyperspheres, Separation of Variables and the Bethe Ansatz J. H A R N A D Department of Mathematics and Statistics, Concordia University 7141 Sherbrooke W., Montreal, Qudbec, Canada H4B IR6, and Centre de recherches math~matiques, Universitd de Montrdal C.P. 6128, succ. centre ville, Montrdal, Quebec, Canada H3C 3J7 e-mail:
[email protected] or
[email protected] and P. W I N T E R N I T Z Centre de recherches math~matiques, Universitd de Montrdal C. P. 6128 succ. centre ville, Montreal, Qudbec, Canada H3C 3J7 e-mail:
[email protected] (Received: 17 May 1994) Abstract. The relation between solutions to Helmholtz's equation on the sphere S "-1 and the [~1(2)]" Gaudin spin chain is clarified. The joint eigenfunctions of the Laplacian and a complete set of commuting second-order operators suggested by the R-matrix approach to integrable systems, based on the loop algebra ~(2)g, are found in terms of homogeneous polynomials in the ambient space. The relation of this method of determining a basis of harmonic functions on S"-~ to the Bethe ansatz approach to integrable systems is explained. Mathematics Subject Classifications (1991). 81R99, 82B23, 17B65, 35J05, 34A26.
O. Introduction In some recent papers on separation of variables in the Hamilton-Jacobi and Schr6dinger equations [1-3], it has been asserted that free motion, both classical and quantum, on the sphere S"-~ and on the negative constant curvature hyperboloid H"-~ is equivalent to the [~1(2)]" Gaudin spin chain [4, 5]. There is, in fact, a close relationship between these systems, which may best be seen through the method of moment map embeddings in loop algebras [6-9], but they actually differ significantly at the quantum level. The purpose of this Letter is, first of all, to clarify what this relationship is and, secondly, to show how an approach that is related to the functional Bethe ansatz [10] for spin chains leads very simply to a basis of harmonic functions on S"-1 generalizing that provided, for the case n = 3, by the Lam6 polynomials [11, 12]. For the separation of variables on S"-1, cf. [13]. Briefly, the difference between the Laplacian on S"- 1 and the [~I(2)]" spin chain system is that, while the former does belong to a commuting family which formally is
62
J. HARNAD AND P. WINTERNITZ
of the same type as the Gaudin systems, the permissible values of the ~I(2) Casimir invariants are different for the two cases. Hence, whereas the classical phase spaces may be simply related, the quantum Hilbert space, joint eigenstates and spectrum are quite different. Moreover, the mapping which determines this correspondence is not one-to-one, but rather involves a quotient by the group Z~ of reflections in the coordinate planes. As a consequence, the Laplacian on S " - ' , together with the associated commuting family of second order operators provided by the loop algebra framework, is diagonalized not on a unique highest weight module of [~1(2)]", as in the Gaudin systems, but rather on the sum of 2" invariant subspaces, each characterized by distinct Z~ transformation properties and containing its own highest weight vector. Essentially, the [~I(2)]" spin chains are defined on tensor products of irreducible discrete series representations of ~1(2), while the Laplacian on S"-~ involves products of the direct sum of the two lowest metaplectic representations. Thus, there are, roughly speaking, 2" times as many harmonic functions on S"-~ as joint eigenstates in an irreducible Gaudin system. In the following, these joint eigenfunctions will be constructed through a separation of variables technique originally developed in the loop algebra approach to classical integrable systems [14a,b]. In the quantum setting, this leads directly to the functional Bethe ansatz eigenstates. The classical case is already amply treated in [7-9, 14a,b], so we proceed directly to the quantum case here. The canonical quantization approach leading to the construction used below is detailed in [15]. Similar formulations may be found in [1, 2, 10, 16].
1. Quantum Moment Map Construction and Commuting Invariants 1.1. ~(2)R LOOP ALGEBRAREPRESENTATION AND COMMUTING INVARIANTS Let (x~,...x,) be the Cartesian coordinates in ~" and (~q,...~t,) a set of n real numbers. Define the operators ei ,=
~-~-7~ 2
xU'
(1.1a) (1.1b)
fi ,= ~xil 2, hi '=
1/
~[xix--,,oxl
(1. lc)
which satisfy the commutation relations [hl, f~] = f l ,
[e,, f , ] = 2hi,
[hl, e i ] = - ei,
(1.2)
and, hence, determine n representations of ~1(2). The constants {#1, ..-#,} are related to the values of the ~1(2) Casimir invariants by h z -- ½(e,~ + f i e , ) = ~ ,u, --
.
(1.3)
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HARMONICS ON HYPERSPHERES
We shall only be concerned here with quadratic combinations of these operators that may consistently be restricted to the unit sphere S n - 1 ~ ~ , . Pick a further set {,1,...~,} of real constants, all distinct, and ordered with increasing values. Define now the following operator-valued rational functions of the 'loop' parameter 2 e(2) := ~
e~
i=i
f(2)
~ --
(1.4a)
~i'
~ fl .= 2 - ai
(1.4b)
hi , 2Zcq
h(2),=a+ ~
i=1
(1.4c)
where a is a real constant. These satisfy the well-known commutation relations for the loop algebra ~(2)R defined with respect to a rational classical R-matrix structure [6, 17]: [hO0, e(#)] =
e(2) - e(#) ){- _- #,
f(£) -f(#)
[h(2), f(#)] -
2- #
(1.Sa)
'
[e(2), f(#)] = - 2 h(2) - h(#) 2-#
(1.5b) (1.Sc)
It follows that the coefficients of the following operator-valued rational function
A(2) := h2(2) - ½(e(2)f(2) + f(2)e(2))
(1.6)
commute amongst themselves: [A(2), A(#)] = 0,
V2, #.
(1.7)
Expanding A(2) in partial fractions gives A(2) = a 2 +
2 ----~i t- K(2) + L2(2) + L'(2),
(1.8)
i=1
where K(2).-=
~
(2--~)2'
i=1
i=1
~t - - (~i
(1.%)
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J. H A R N A D
A N D P. W I N T E R N I T Z
and {Hi}i= 1 ..... are a set of commuting second-order differential operators defined by
- L ~ + #2x~ + #j
1 Hi:--~
+ a xi
i
(~i - - ~ j
j=l
(1.10)
+ ~X/
'
j~.i
where (1.11)
Lij:= xi-- - xj ~xj Ox~
is the generator of rotations in the (ij) plane. For a H 0, the Hds are linearly independent and their sum is H~=a
D+
(1.12)
,
/=1
where D-'= i= 1
(1.13)
xi - ~Xi
is the Euler homogeneity operator. For a = 0, this sum vanishes, so only n - 1 of the operators are independent. But since they still all commute with the Euler operator D, this can be adjoined as the nth independent commuting invariant in this case. Contained within this framework, for particular values of the constants {a, #1,... #,}, are both the Laplacian on the sphere S "- 1 and the Gaudin [~I(2)]" spin chains. It is important to note, however, that they correspond to different values of the Casimir invariants for the various ~l(2) representations. To obtain the Laplacian on S ~- 1, we set a and all the #/'s equal to zero. Since the Hds are defined in terms of the rotation generators L~j, they may be restricted consistently to S n- 1. Forming the sum Ho ,= 4
aiHi = -- ~ i=1
(1.14)
L2 t
1
and restricting to the unit sphere S ~- 1 gives the Laplacian H o l s , - 1 = - As,-1.
(1.15)
More generally, if the #/'s are nonvanishing, we have H
aiHi = -- ~
.'= 4 i=1
i,j=l
i=1
j=l
~.2 -Xj
i=1
and restricting to S n- 1 gives H Is. . . . .
As._, +
x2 -i=1
#~. i=1
(1.17)
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H A R M O N I C S O N HYPERSPHERES
This corresponds to a degenerate case of the quantum Rosochatius system [15, 18], in which the harmonic oscillator potential is absent. (It may also be obtained as a reduction of the free system on a sphere of dimension 2n - 1 under the action of the maximal torus in the isometry group SO(2n) (cf. [19]).) The other operators Hi in the commuting family may similarly be restricted to define commuting operators on S"- 1.
1.2. RELATION TO THE [~1(2)]" G A U D I N CHAIN
To obtain the [~I(2)]" Gaudin spin chain, we must make a change of representation. Instead of considering square-integrable functions on N" or S "- 1, we must reinterpret the operators {ei, fi, hi} entering in the definitions (1.4a)-(l.4c), (1.6) as acting upon a highest weight module of [~I(2)]" of the type = 6
9ffh,
(1.18)
i=1
where the individual factors ~z, in the tensor product are generated by application of the ladder operator fi to a unique highest weight vector [O>t,, the kernel of the operator e~. (We make no notational distinction between an operator (9 acting on ~ h and its extension I ® I ® ..- ® (9 ®--. ® I acting on ~.) Note that the space of square-integrable smooth functions on R", with the representation (1.1a)-(1.1c), will not do for this purpose, since the kernel of the operator (1.1a) is two-dimensional. However, ignoring for the present the question of normalizability, we may still consider the space obtained by application of polynomials in the operators f~ to any specifically chosen vector ]O>z, within the kernel of each ei. This kernel is of the form ~ i = {aixfli i -~- b i x ; (pl- 1)},
(1.19)
fli(fl,- 1) = #z.
(1.20)
where
For ~ti # 0, we may choose the unique positive root of this quadratic equation and identify, up to normalization [0>h ~ x ~'.
(1.21)
If the domain of definition is ~", to obtain single-valued functions, the /3~'s must be chosen to have integer values. Then ~ t , consists of functions of the form p~ p(xz2 )xi, where p is a polynomial in x~2, and ~ consists of functions of the form ~(x~,.. . x ,2) H i,= l x f ' , where ~ is a polynomial in its n arguments {xZ,...x2}. In the case where the ~i's vanish, however, which is the one relevant to the Laplacian on S"-1, each of the fli's may have the two possible values
fli=O
or
fli=l,
(1.22)
and we obtain distinct highest weight representations for each choice of the fl[s.
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J. H A R N A D A N D P. W I N T E R N I T Z
To obtain the ~l(2) analogue of the highest weight representations used in [10], we must conjugate the operators {ei, f~, hl} by the factor x(' and re-express the result in terms of the new variable (1.23)
Z i : = 1. 2Xi 2 •
Thus, we define 62 ei '= xT~ P'eix( ' = Zi ~z ~ + 2li ~3z--~i'
(1.24a)
fl '= x[P'fix~ ' = zi,
(1.24b)
O hi := x ; a ' h i x ~ ' = Zi~z ~ + li,
(1.24c)
where fli 1 li .'=~- + ~.
(1.25)
This is the ~1(2) analogue of the representation used in [10] to describe the ~u(2) Gaudin spin chain in functional terms (cf. [20, 21]). The highest weight state 10> in this representation is just a constant, and a basis for the corresponding space o@ generated by application of the f ' s is given by the homogeneous polynomials in the variables {z 1,... z, }. Within such representations, the fli's need not have integer values. For any set of positive values for the li's, a scalar product may be defined by j° k, k ¢, , a=l
(2.27)
10; fll .... , ft,> .'= ly] x(' i=1
(2.28)
where
denotes the highest weight vector in the Z~-invariant subspace ~P(fll,..., ft,). Summing over the eigenvalues of the H~'s to obtain that of H in (1.16) gives 4 =1 ~n ~,E~= ( 2p + i fli ) ( 2p + ~ fl, + n "= i=1 i=1 + 4a
Vb +
i=1
Cq(fli + ½) •
2 ) + ~ (fl~_ fl2) + i=1 (2.29)
For the case when a = 0 and each fl~ is either 0 or 1, this reduces to the usual formula [24] for the eigenvalues E,,,t of the Laplacian on S"-1 E,,z = - l(1 + n - 2),
(2.30)
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J. H A R N A D A N D P. W I N T E R N I T Z
expressed in terms of the total degree of homogeneity
l..= 2p + ~, fl~.
(2.31)
i=1
To count the total number of eigenfunctions in this case with a given degree of homogeneity l, let e = 0 or 1, for l even or odd, respectively, and
m:= min([~-~], [~]).
(2.32)
Summing over the number of eigenfunctions for each combination of values of p and fl~'s giving the same homogeneity l, the total number is a~,l= ~ ( n + [ / / 2 ] - 2 k:O n-- 2
-k)(
n )=(n+2l-2)(n+l-3)[ 2k + e l!(n- 2)!
(2.33) '
which is the number of homogeneous polynomials of degree l which may not be obtained by multiplying those of degree l - 2 by r 2. This agrees with the usual count for the number of harmonic functions on S "-~ obtained, e.g., through the use of Gel'fand-Tseitlin bases [24].
3. Conclusions and Discussion We have seen that a basis of harmonic functions on S "- t may be obtained in terms of homogeneous polynomials that are simultaneous eigenfunctions of the complete set of commuting second-order operators {H1,..., Hn} provided by the ~(2)R loop algebra framework, as well as of the generators of the group Z ~ of reflections in the coordinate planes. This gives a natural generalization of the harmonic functions on S 2 provided by the Lam6 polynomials [11, 12]. More generally, the same method determines the joint eigenfunctions of the systems associated to the operator (1.16) for #~ = fli(fli- 1) with fli's any nonnegative integers. These eigenfunctions provide bases for the spaces ~tf = ®i= ~ ~ h formed from metaplectic representations of d(2) with highest weights li = flJ2+¼. Since these are just the functional Bethe ansatz eigenstates for the [d(2)] n Gaudin spin chain within these representations, their completeness is equivalent to the completeness of the Bethe ansatz for this case. It may be worthwhile noting that the decomposition of the relevant Hilbert space into the 2 n irreducible subspaces formed from tensor products of metaplectic representations of the type [(¼) • (¼)]n suggests the presence of a supersymmetric structure involving representations of o~p(2, 1) (cf. [25]). Gaudin spin chains based on such superalgebras have been considered recently [26"] within the Bethe ansatz approach. Extending the functional formulation to such superalgebras may serve to further clarify the r61e of the Z~ invariance of the Laplacian and the associated commuting operators on S ~-t, as well as to prove the completeness of the Bethe ansatz eigenstates in the supersymmetric case.
HARMONICS ON HYPERSPHERES
73
The method of solution used here may also be extended to other commuting operators on S "-1 arising from the ~(2)R loop algebra approach, including, e.g., harmonic oscillator interactions [,-15,18, 273 and modifications of the #2/x2 interaction terms in (1.17) due to degeneracies in the space of parameters {~1,...~,} [-3, 28]. However, since the homogeneity operator D does not necessarily commute with the operators in question, the joint eigenfunctions may no longer simply be homogeneous polynomials; in fact, they need not be polynomials at all. These questions and further extensions of the loop algebra separation of variables method will be addressed in future work.
Acknowledgements The authors have benefitted from helpful discussions with A. el Gradechi and E. G. Kalnins on matters related to this work. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada and the Fonds FCAR du Qu6bec.
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