HARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE

We extend the spectral analysis of differential forms on the disk (viewed as the non-Euclidean plane) in recent work by J. Peetre L. Peng G. Zhang to the dual situation of the Riemann sphere S2. In particular, we determine a concrete orthogonal base in the relevant Hilbert space L,2($2), where is the degree of the form, a section of a certain holomorphic line bundle over the sphere S2. It turns out that the eigenvalue problem of the corresponding invariant Laplacean is equivalent to an infinite system of one dimensional Schr6dinger operators. They correspond to the Morse potential in the case of the disk. In the course of the discussion mny special functions (hypergeometric functions, orthogonal polynomials etc.) come up. We give also an application to "Ha-plitz" theory.

The plan of the paper is as follows.In Section 1 we uncover the structure of the eigenspaces of the invariant Laplacean, in particular we find an explicit orthogonal basis in them.A novelty is perhaps that we determine the spectrum factoring suitable functions of the Laplacean.In fact, the same thing can be done on any Riemann surface so we have deferred the details of the proof to an appendix (as a Section 6).In Section 2 we use the orthogonal basis from Section 1 to write down the reproducing kernels involved.Some applications in the spirit of Vilenkin's great book [12] are given in Section 3.Many more such applications are possible!In Section 4 some operator theory consequences of the previous discussion are briefly indicated.Lastly, in Section 5 we return to the basis vectors and consider the differential equations satisfied by them.In particular, we write down the "periodic" analogue of the Morse operator.
We shall work within a fixed coordinate neighborhood with coordinate z obtained by deleting one point cx ("the point at infinity").In other words, S 2 will be identified with the extended complex plane : CLJ {cx} (or the complex projective line I1).Near cx we use instead of z the 1 coordinate The essential change compared to [10] is that the factor 1 -Izl 2, everywhere, has to be replaced by 1 + izl (nd X z by 1 + zt). 2 Thus, the invariant Cauchy-Riemann operator is given by O (1 + Izl) 0 Similarly, the invariant Laplacean is A, --(1 + Izl) 02 ,9 0zO5 + ,(1 + Izl)' b-7 The relevant group is now the compact group SU(2), consisting of all 2 x 2 complex matrices =(ac d b) such that ad-bc l' c -' d ' and the actin we have in mind is U).f(zf((z))(t(z))_.f(az + b z + d )(z + d)" In this notation we have DU () U(+2)D.
Notice that, compared to [10], we have changed r, to -r, in the last formula.Also we are going to assume that r, is an integer > 0. This is very convenient, as we are going to work with the Hilbert space L2'u(S2) of functions with the metric da(z) Ilfll If(z)12(1 + Izl)"+ ' where da is the normalized area measure, da(z) dxdy.da(z) REMARK.The measure ( + [z[2) '+ is sometimes named after various authors: Berezin, Berg- man, Dzhrabshyan, ttarish-Chandra, Kostant etc. lom a highbrow point of view we should, strictly speaking, consider the elements of this space not as functions but as sections of an Hermitean holomorphic line bundle over ,5 '2 (differential forms of degree -).In fancy language, the Picard group of p1 is isomorphic to/', so the holomorphic line bundles over P1 are labeled by a discrete parameter ,.If we interpret the formula w az + b cz + d as a change of coordinates on P, the family {(cz + d)"} gives the transition functions.
For details see the Appendix, where the proof is given in the context of an arbitrary Riemann surface equipped with a metric with positive Gaussian curvature.Now we begin to uncover the structure of the space Bt.where each gj 0 O, 1,..., 1) is a polynomiM of degree <_ u + 21.However, the -j highest coecients of 9j are determined by 9,..., 9+1.In particular, the dimension of Bt is + 1)( + +).
PROOF: That every lution of Dt+ has the representation (1) with 9 entire in z follows in [10] relying the differential equation.In order to see which are the restrictions on 9 at we 1 me the ehge of able z Then we find For instance, the l-th coecient in the outer sum is, up to sign, just gt()z +.It foows that gt must be a polynomial of degr v + 21.Similarly, the (l-1)-st coefficient is, again up to sign, I-1 )zU+21_ 9'(1+ + '-'(7- It follows that gt-must be a polynomial of degree _< v+2/-1 and that llt(v+21)+.lt_(v+21-1) 0. (We use the sign to denote Taylor coefficients.)The general statement about the functions gj follows now readily by induction.It is now likewise clear that dimBt =v + 21+ 1 + (v + 21-2 + 1) +..-+ (v + 2 + 1) + (v + 0 + 1) =(1 + 1)(v + 1) + 2(1+ Next, we decompose the space Bt under the action of the rotation group SO(2) (the isotropy group of the origin z 0).The elements of the nth space in this decomposition rhust be of the form {z[ 2 (2) f =z"q(), where q q(t) is a polynomial of degree _< 1.If n < 0 it must vanish to the order -n at the origin 0: q(t) O(t-").Similarly, if n > v it must vanish to the order n-v at the point 1 1: q(t) O((1 t)"-u).The last statement follows again by making the substitution z -Z We then get (3) f '-"q(1 X / I1 )" It follows that we must have -I < n < + v.
{{QI{ {Q(t,){2(1 + t: (1 t, Thus if 0 < n < v we have the conventional Jacobi polynomials P"'v-") (up to normalizing factors).If we agree to use the same notation also in the general case, we can write our basis as lzl+ 1 }' where as before -l < n _< + u.A possibly even better parameter than t is t2 1 t 1 2t 1 -Izl If we realize S as the standard sphere in N (using stereographic projection onto ), 1 / Izl " it gets an immediate geometric meaning.Namely, if we let 9, ff be polar coordinates about the north pole, we can write z tan : e '*.It follows then that, indeed, t cos/9.
It should be clear that our result generalizes the classical Laplace series (expansion in spherical harmonics).This is the case u 0, when the elements are genuine functions, not sections of a line bundle.For instance, it is well-known that the Ith eigenvalue of the Laplace-Beltrami operator equals l(l + 1), which agrees with our formula.
where X is a function of one variable.

else
This is a mtiplication threm.
Next we invoke the trsvectt (s [4], [15]).It is question of the following bi]ine differenti expression" If we let fl transform with weight vl and f2 transform with weight ix2, then the transvectant 7i(fl, f2) transforms with weight ix1 + ix + 2s.
2,, A2,+2/ We can now exhibit an isomorphism between the spaces A and which respects the SV(2)-action (an "intertwining" map).Namely, we take in (8) ix -ix-21, v2 1, :f g E A ''+2t .[K_(z,z) (1 + Izl2) - This gives an element f E A 'u defined by To see that the map g f indeed is an isomorphism we take g z"+t, -1 _< n _< v + I.This gives us, apart from a factor, back the basis vector e,t (use Lemma 4): e,t(z).I Let {tnm()} be the matrix of action , on the space in terms of the basis {z"+t}, -l < n < n-l, i.The functions tnrn() are expressible in terms of Legendre functions (cf.[12]).Using the isomor- 2, phism constructed above we see that the action of v) on A has the same matrix: As a generalization of (10), we may, following the procedure in [15], consider for an integer s the operators T( ') with kernel ( T(t)(z, w)= (1 + zt)" z where we agn sume that the symbol B trfforms with weight -t.If a 0, (12) reduces to (10).In other words: TB T -').The connection betwn a, s d in generM will be uncovered now.
As there is no derengagion in the w vable it is agn tfiviM ha we have the correc w behavior.We require that the product with brackets in (12) be of weight 1 .This gives the relation (13) s-t-t/'=l-o or o+s=t+v'+l.
If we apply "Bol's lemma" (see [4]) it follows that, after the differentation carried out, we have an expression whose weigth is 1 + o.For the result to have the required weight -t/we then get an equation which we may write as (14) s--t/=l+a or a-s=-v-1.
To see this, we observe that if we differentiate the factor (1 + ztO)'-" he exponent goes down to v' s a. From this we obtain v' a > 0, which is in view of (15) the same as (17).| Thus altogether we have (( 16) + ( 17)) (18) t/--<'-< + This agrees with the classical fact that A2,,, @ A u' AZ,t where we sum over precisely the indices occuring in (1) with the same parity as the difference v-t/' (see [12], page 177).Thus we have made this decomposition rather explicit.Note that the range of a is the interval [0, v'].
Let us indicate one application of the preceding connected with Clebsch-Gordan coefficients.
It follows that the operator intertwines with the SO(2)-action: t, ),/, ,7,( where U#f(z)= f(e'z).This again implies that monomials are mapped onto monomials.More precisely, we have n' ,.z =Ct(n,n')z" with n=n +k-a, where the numbers Ct(n,n') are our version of the Clebsch-Gordan coefficients. (We do not indicate in the notation the dependence on v and v'.) For a general symbol this gives t)(n,n') Ct(n,n')(n n' + a), where we presently use the hat to denote Fourier coefficients.Now a general operator can, by the above, be decomposed into a sum of Toeplitz type operators: It follows that we have the following expression for the Hilbert-Schmidt (S2-)norm: n=0 n'=0 In particular, we obtain the following orthogonality relation (cf.[9] for a similar result in the hyperbolic case): =const C,(n,n')C,,(n,n')llz"ll2llz"'ll2,o if # t'.

DIFFERENTIAL EQUATIONS SATISFIED BY THE BASIS VECTORS.
In Section 1 we constructed the basis {e,a()} *"q"(i I1 ) where -l _< n _< + v, in the space L,'(S).Now we write down various ordinary differential equations connected with these functions.
n+l --(u+2)t This is of course just a special case of the hypergeometric equation in slight disguise.
On the other hand, we know by their very construction that the functions e,t(z) satisfy the partial differential equation (notation: 0 zz' 0 ).
It is very curious that essentially the same differential equation in two slightly different guises appeared in the same year 1929 s mad in two completely different fields: in statistics, respectively in quantum theory.
Let us formally work out the connection.
To swer this question we must first digress a little.DIGRESSION: DUCTION OF A SELF-ADJOINT SECOND ORDER OINARY DIFFER- ENTIAL EQUATIONS TO NODAL FORM.
Consider a -adjoint eigenvMue equation of the form u_w_d dq] wa +bq=q in L2(I, wdx), where I is some intervM C d w is a positive weight.Furtheore, we sume that the coecient a is positive and, similly, that b is reM.So the operator is foy seg- adjoint d we expect it to be sebounded from below on a stable domMn.We wish to reduce it to the normM -f" + in Le(J, dx), where J is some other intevM (the ce a w 1).We will thi of V the
PROPOSITION 2. The eigenvMue problem for the invariant Laplace operator for -v-forms on S 2 is equivadent, in a sense made precise in the foregoing, to an infinite system of synchronous Schrfd/nger (or Sturm-Liouville) eigenva/ue equations on the interva/.(0,r).| REMARK.Why is there only one equation in the hyperbolic case?It is because in [10] we used the halfspace realization of non-Euclidean geometry, so we have a dilation invariant situation, which thus accounts for the "degeneracy".If we were to use the disk realization, we would certianly again get infinitely many equations.(One has only to substitute, in the above formula, the trigonometric functions by their hyperbolic ("alcoholic") counterparts tanh and coth .)We do not know if any of this has any bearing to physics whatsoever.
6. APPENDIX.FACTORIZATIONS OF CERTAIN DIFFERENTIAL OPERATORS ON A RIEMANN SURFACE.
We continue the computation in [10], remark in Section 3.
As there, we let X denote a Riemann surface equipped with an Hermitean metric, in terms of a local coordinate z given by ds 2 (z)ldzl .Wec onsider t,-forms locally given in a coordinate neighborhood (with coordinate z) by f f(z)(dz)".If we make a change of coordinate (z X(z)) then these coefficients experience the change f(z) ,--+ f(X(z))(X'(z))", 9(z) g(x(z))lx'(z)l 2.
(Notice that the parameter v plays in this Section the same rrle as v/2 in (the rest of) [10], while compared to the rest of the present paper it is the same as -v/2.)We denote by L2'"(X) the space of square integrable v-forms, that is, f E L2'(S2) if and only if fx 0 0 Let us write 0 z' (Wirtinger operators).The invariant Cauchy-Riemann operator is given by D -'.
We further set D,, -0 + (r, 1) Og g It may be viewed as the adjoint of/) =/),, regarded as an operator from L 2,u into L2'-1.In particular, it maps L 2,-' into L 2,.(In [10] we interpreted D as the "metric" connection on the sheaf of all holomorphic v-forms.)The corresponding Laplace operator is defined by A A,, D,,/).
In [10] we had, for some reason, written the factors in a different order.We now explain this discrepancy better.
From now on we assume that we have a metric with constant curvature K.We claim that one can define recursively polynomials Or(T) (where the letter T stands for an indeterminate) such that D,,D,,_I D,,_D +a Ot(A).
Indeed, if 0 this is just the definition of A A, with O0(T) T. Assume that Ot_l is already defined and multiply the corresponding relation (l-1 instead of l!) with A from the right.This gives: D,,D,_I ...D,_(#_I)DD,D O-I(A) A.
Altogether, we have now established the following result.THEOREM 2. We have the following factorization D,,D,_... D,_,/) TM A(A + (2 2)-) (A + 2(2v 3)-)... (A +./(2vI -l)).| We conclude with several remarks.REMARK.Introducing the graded vector space L ]e L2'u, where the summation is over some remainder class of the indices u mod Z, we can regard the operator /) as an endomorphism of L and, similarly, we can define an endomorphism D on L extending D, L 2'" L2'"+1.Then one can get a more elegant formulation of the above results.In particular, we can write D,D,_...D,,_D + more compactly just as D+/)+.This is in line with how it is done in cohomology theory.REMARK.To some extent the purported generality of an arbitrary Riemann surface is illusory, as by the uniformization theorem one can reduce oneself to the situation of a simply connected manifold, that is, either the disk, the disk or the ("parabolic") plane.One can then always assume that the metric is given by 1 g- (1 + K--) 2"   It follows that