Irsee Talk.ps

The Cheshire Cat ee t in Lie theory: Variable separation and superintegrable systems W. Miller, S hool of Mathemati s, University of Minnesota, Minneapolis, Minnesota, 55455, U.S.A. millerima.umn.edu June 16, 2004

1 Abstra t There are many physi ally important partial dierential equations in quantum and lassi al me hani s that admit Lie symmetry groups, but for whi h the symmetry is destroyed by addition of symmetry breaking potentials or

onsideration of non-symmetri solutions. For superintegrable systems on

onstant urvature spa es there is a Cheshire Cat ee t: the symmetry is gone but its in uen e lingers on and an be used to analyze the solution spa es. We des ribe this general approa h to harmoni analysis and give a number of examples. In parti ular we show how this approa h leads to new results for Lame' and Heun fun tions, and insight into the lassi al Niven transform. Joint work with Ernie Kalnins, Jonathan Kress, George Pogosyan and Vadim Kuznetsov.

1

2 Introdu tion Most fun tions ommonly alled \ spe ial " obey symmetry properties that are best des ribed via group theory (the mathemati s of symmetry. We have in mind: 1. Spe ial fun tions as matrix elements of Lie group representations. (addition theorems, orthogonality relations, spheri al fun tions) 2. Spe ial fun tions as basis fun tions for Lie group representations (generating fun tions) 3. Spe ial fun tions as Clebs h-Gordan oeÆ ients for the redu tion of tensor produ ts of irredu ible group representations (the motivation for Wilson polynomials). There are, however, many physi ally important partial dierential equations in quantum and lassi al me hani s that admit Lie symmetry groups, but for whi h the symmetry is destroyed by addition of symmetry breaking potentials or onsideration of non-symmetri solutions. We will show that for superintegrable systems on onstant urvature spa es there is a Cheshire Cat ee t: the symmetry is gone but its in uen e lingers on and an be used to analyze the solution spa es. Our emphasis will be on 1. Spe ial fun tions as solutions of Lapla e-Beltrami eigenvalue problems on a Riemannian manifold (with potential) via separation of variables. There is no general onne tion between the symmetry group and algebra of a manifold and the separable oordinate systems. However for onstant urvature spa es this onne tion does exist, at the level of se ond-order elements in the universal enveloping algebra of the symmetry Lie algebra, and it may persist even when the symmetry is ompletely broken by a potential. We will des ribe these relations and give appli ations to spe ial fun tions, su h as Lame' and Heun fun tions, that dire tly have no Lie symmetry properties.

2

3 Symmetries of dierential equations Let D be a linear partial dierential operator in n dimensions (with lo ally analyti oeÆ ients). Let  be a parameter.

De nition 1 The linear partial dierential operator S is a symmetry operator for the equation D =  if S maps lo al solutions  to lo al solutions S . This is basi ally equivalent to the requirement that [S; D℄ = 0. The linear partial dierential operator S~ is a onformal symmetry operator for the equation D = 0 if S~ maps lo al solutions  of D = 0 to lo al solutions S . The rst order symmetry operators for D =  form a Lie algebra, the symmetry algebra of this equation. Indeed, if L1 ; L2 are rst order symmetry operators, then the ommutator [L1 ; L1 ℄ is a rst order symmetry operator. The asso iated lo al Lie symmetry group maps solutions to solutions. The rst order onformal symmetry operators for D = 0 form a Lie algebra, the onformal symmetry algebra of this equation. The asso iated lo al Lie onformal symmetry group maps solutions to solutions.

3

4 Intrinsi hara terization of variable separation Let Vn be an n-dimensional Riemannian manifold and V be a potential fun tion on Vn . The Lapla e-Beltrami eigenvalue equation (with potential), or S hrodinger equation for fun tions on Vn is

H (q) = E (q)

(1)

where in lo al oordinates fq ` g

H =
n + V (q);

n  p jk  1 X
( gg ) qk
n = p g j;k=1 q j

(2)

and g = det(g jk ) 1 . The Lapla e equation is

H (q) = 0:

(3)

A parti ular set of orthogonal oordinates fx` g is R-separable for (1) if this equation admits solutions of the form = exp(R(x))ni=1 i (xi ) = eR ; where R(x) is a xed fun tion, independent of parameters, and the fa tors i (xi ) are the solutions of n ODEs (the separation equations) 0

00i + gi (xi ) 0i

fi (xi ) +



1

n

X

j =1

j sij (xi )A i = 0; i = 1;


; n

(4)

and 1 = E . The parameters j are the separation onstants. If R  0 Pn we have separation, and if R(x) = i=1 R(i) (xi ) we have trivial R-separation. Otherwise the R-separation is nontrivial. There is a orresponding de nition of R-separation for the Lapla e equation (3) with E = 0. The n  n matrix (the Sta kel matrix) is required to be nonsingular.

4

The basi result in the theory is

Theorem 1 Ne essary and suÆ ient onditions for the existen e of an orthogonal R-separable oordinate system fxi g for the S hrodinger equation (
n + V ) = E on an n-dimensional Riemannian manifold are that there exists a linearly independent set fS1 = H =
n + V; S2 ;


; Sng of se ondorder partial dierential operators on the manifold su h that: 1. [Sk ; S`℄ = 0; 1  k; `  n, 2. Ea h Sh is in self-adjoint form (in any lo al oordinate system): n 1 X  p jk ` 
( ga(h) (q )) qk + W(h) (q`); ajk(h) = akj(h) S` = p g j;k=1 q j

3. There is a basis f!(j ) : 1  j  ng of simultaneous eigenforms for the fajk(h) g. If onditions (1)-(3) are satis ed then there exist fun tions gi (q) su h that:

!(j ) = gj dxj ; j = 1;


; n; i.e., the eigenforms are proportional to the separable oordinates.

Corollary 1 If onditions (1)-(3) are satisi ed then the R-separable solutions 1 ;


;n (x) = exp(R(x))ni=1 i (xi ) are hara terized as the simultaneous eigenfun tions of the ommuting symmetry operators Sh : Sh 1 ;


;n = h 1 ;


;n ; h = 1;


; n

5

REMARKS: 1. No ne essary onne tion between the se ond-order dierential symmetry operators Sj that des ribe separation on Vn (plus a potential V (q) and the Lie algebra of rst-order symmetries of Vn . However, if Vn is a spa e of onstant urvature and V (q)  0 then all higher order symmetries of this spa e an be onsidered as elements of the universal enveloping algebra of the Lie symmetry algebra Gn of the manifold: the Eu lidean Lie algebra e(n; R) for Eu lidean spa e, so(n; R) for the n-sphere, et . Similarly The onformal symmetry algebra for the Eu lidian spa e Lapla e equation is so(n; 2). 2. If Vn is a spa e of onstant urvature and V (q) 6= 0 then the symmetry operators (or onformal symmetry operators) des ribing variable separation take the form Sj = Sj + Wj where Sj is a se ond-order element in the enveloping algebra of Gn and Wj is a s alar (potential) fun tion. 3. For onstant urvature spa es there is no nontrival R-separation for the equation H = E , i.e., we an set R = 0 and only onsider pure separation . However, nontrivial R-separation does o

ur for the Lapa e equation H = 0. 4. There is a natural ( oordinate independent) inner produ t on the spa e L2 (Vn ) of square integrable fun tions F; G; on Vn , given by

< F; G >=

Z

p

F (q)G(q) g dq:

The se ond-order symmetry operators Sj are formally symmetri with respe t to this inner produ t: < S` F; G >=< F; S` G >.

6

How does one nd all orthogonal separable oordinate systems q of H = E for a given spa e Vn for zero potential, V  0? This is a diÆ ult problem in dierential geometry. For real n-dimensional Eu lidean spa e, the n-sphere, and the n-hyperboloid of two sheets, Kalnins and the author have a graphi al pro edure to lassify and onstru t all possibilities. On e the zero potential separation problem is solved, it is straightforward to ompute the possible separable systems for any nonzero potential, as a restri tion of zero-potential separable systems. In real Eu lidean 3-spa e there are 11 separable systems see Table 1. Here the equation H = E is (X2 + Y2 + Z2 ) = ! 2 : A basis for the symmetry algebra is : PX ; PY ; PZ ; MZY ; MXZ ; MY X where

PX = X ;


MZY = ZY

Y Z = MY Z ;


;

the ommutation relations are [MY Z ; MZX ℄ = MY X ; [MZY ; PY ℄ = PZ ; [PX ; PY ℄ = 0 and the modi ations of these obtained by y li ally permuting X; Y; Z . We de ne 2 2 P
P = PX2 + PY2 + PZ2 ; M
M = MZY + MXZ + MY2 Z :

For the Lapla e equation (X2 + Y2 + Z2 ) = 0 orthogonal separation is possible in the 11 Helmholtz separable systems and nontrivial R- separation in 6 additional systems.

7

5 Hilbert spa e models for the solution spa e of the Helmoltz equation We return to the 3D Eu lidean spa e Helmholtz equation with zero potential: (X2 + Y2 + Z2 ) = ! 2 : We an impose a Hilbert spa e stru ture on the solution spa e of this equation su h that the operators de ning variable separation are self-adjoint. This is done by taking the Fourier transform. Consider the mapping I from fun tions on the two-sphere to the solution spa e: (X) =

Z Z

S2

ei!X
K h(K)d (K) = I (h);

X = (X; Y; Z ); K = (k1 ; k2; k3 ); K
K = 1: Here K is a unit ve tor that runs over the unit two-sphere S2 , d is the usual solid-angle measure on the sphere. The set L2 (S2 ) of measurable fun tions h is a Hilbert spa e with inner produ t < h1 ; h2 >=

Z Z

S2

h1 (K)h2 (K)d (K):

In spheri al oordinates on S2 we have K = (sin  os ; sin  sin ; os ); d (K) = sin  d d: The Lie algebra generators on L2 (S2 ) indu ed by the symmetries on the solution spa e of the Helmholtz equations are PX = i!k1; PY = i!k2; PX = i!k3; k12 + k22 + k32 = 1; MZY = k3 k2 k2k3 = sin  + os  ot ; MXZ = k1k3 k3k1 = os  +sin  ot ; MY X = k2 k1 k1k2 =  : We an use the two-variable model L2 (S2 ) to work out the pre ise spe tral resolutions for pairs of formally ommuting symmetry operators S; S 0 that de ne a separation of variables for the Helmholtz equation. That is, we

ompute a basis of eigenfun tions: S f = f ; S 0 f = f ; < f ; f  >= Æ( 0)Æ( 0): Then the fun tions  (X) = I (f ) will form a orresponding basis on the solution spa e of the Helmholtz equation. 0

8

0

Table 1 Separable oordinates in 3-D real Eu lidean spa e. Coordinate System

Coordinates and Operators S1 ; S2

I. Cartesian x; y; z 2 R II. Cylindri al polar  > 0, ' 2 [0; 2 ) III. Cylindri al ellipti z 2 R, d > 0 IV. Cylindri al paraboli ; x 2 R,   0 V. Spheri al r > 0;  2 [0;  ℄, ' 2 [0; 2 ) VI. Prolate spheroidal a > 0, ' 2 [0; 2 ) VII. Oblate spheroidal a > 0, ' 2 [0; 2 ) VIII. Sphero- oni al r0 0 < b < 1 < 2 < 1b ; 0 < 1 < 1 IX. Paraboli ;   0, ' 2 [0; 2 )

x; y; z S1 = PX2 ; S2 = PY2 x =  os ', y =  sin ', z S1 = MY2 X ; S2 = PZ2 x = d osh  os , y = d sinh  sin , z S1 = MY2 X + d2 PX2 ; S2 = PZ2 z , y =  , x = 21 ( 2  2 ) S1 = MY X PY + PY MY X; S2 = PZ2 x = r sin  os ', y = r sin  sin ', z = r os  S1 = M
M; S2 = MY2 X x = a sinh  sin  os ', y = a sinh  sin  sin ', z = a osh  os ' S1 = M
M a2 (PX2 + PY2 ); S2 = MY2 X x = a osh  sin  os ', y = a osh  sin  sin ' z = a sinh  os ' S1 = M
M + a2 (PX2 + PY2 ); S2 = MY2 X b2 1) , y 2 = r2 b(1 b1)(12 1) x2 = r2 (b1 1)( 1 b z 2 = r2 b1 2 2 2 S1 = M
M; S2 = MZY + bMXZ x =  os ', y =  sin ', z = 12 ( 2  2 ) S1 = MZY PY + PY MZY MXZ PX + PX MXZ , S2 = MY2 X x2 = (u1 a)(au(a2 1)a)(u3 a) , y 2 = (u1 1)(u12 a1)(u3 1) z 2 = u1 ua2 u3 S1 = PX2 + aPY2 + (a + 1)PZ2 + M
M, 2 2 S2 = MXZ + a(MZY + PZ2 ) x = 2 osh  os sinh , y = 2 sinh  sin osh z = 2 ( osh 2 + os 2 osh 2 ) S1 = MY2 X 2 PZ2 +

(MXZ PX + PX MXZ + MZY PY + PY MZY ) S2 = (PY2 PX2 )+ MXZ PX + PX MXZ MZY PY PY MZY 9

X. Ellipsoidal 0 < u1 < 1 < u2 < a < u3 XI. Paraboloidal 0 < ; o  ; ' 2 [0; 2 )

Examples: 1. Spheri al system.

fm(`) (; ) = Y`m (; ); (m`) (X) = 4i` j` (!r)Y`m (; );

m = 0; 1;


; `; ` = 0; 1;


S

1

= `(` + 1)E; S2 = m2 E:

Here, < fm(`) ; fm(` ) >= Æmm Æ`` , Y`m is a spheri al harmoni and j` is a spheri al Bessel fun tion. 0

0

0

0

2. Paraboli system.

fm (; ) =

p

[tan(=2)℄ 2 sin 

i

eim ;

im 2 1 m + i 1 m i e i=2 ! 2 ;m (X) = ℄ ( ) ( )Mi=2; m=2 [ p ! 2 2 2 i=2 2 Mi=2; m=2 [ e p! ℄eim ; 2 m = 0; 1;


1 <  < 1 S1 = 2!E; S2 = m2 E: Here, < f;m ; f ;m >= Æ  Æmm , and M; is a Whittaker fun tion. 0

0

0

0

3. Ellipsoidal system. m m m m m elpn m (; ) = elm n ( )eln ( ); nm (X)) = Kn (!; k )eln ()eln ( )eln ( );

m = 0; 1;


; n n = 0; 1;


S

= mn E; S2 = mn E: Here, < elpn m ; elpn m >= Ænn Æmm , and elm n is an ellipsoidal wave fun 1=2 tion with modulus k = a . 1

0

0

0

0

The expressions = I (f ) are nontrivial integral representations for the separated solutions. However, the integrals are easy to evaluate be ause we know in advan e that they separate in the orresponding oordinates. Thus we need only al ulate normalization onstants.

10

6 Hilbert spa e expansion formulas To expand one type of separable basis in terms of another, it is most onvenient to ompute the expansion oeÆ ients in the sphere model. For example, the overlap integrals between the spheri al basis and the paraboli basis an be shown to be

< fm(`) ; f;m >= Æmm 0

v

m+jmj u

0

( 1) 2 (jmj!)2

u t

(2` + 1)(` + jmj)! i + jmj + 1 ( ) 4 (` jmj)! 2

j `; jmj + ` + 1; i+j2mj+1 j1 ;  ( i + 2jmj + 1 )3F2 jjm mj + 1 ; j mj + 1 m = 0; 1;


; `. This allows us the expand a spheri al basis fun tion as an integral over produ ts of Whittaker fun tions. !

11

7 Conformal symmetries of the Lapla e equation and the lower-variable models Basi fa ts on erning the solution spa e of the 3-variable Lapla e equation in Eu lidean spa e (X2 + Y2 + Z2 ) = 0: (5) The onformal symmetry algebra of this equation is 10-dimensional, with basis

PX = X ; PY = Y ; PZ = Z ; MY X = MXY = Y X

XY ;

1 MXZ = ZZ ZX ; MZY = ZY Y Z ; D = ( + XX + Y Y + ZZ ); 2 2 2 KX = 2XD R X ; KY = 2Y D R Y ; KZ = 2ZD R2 Z ; where R2 = X 2 + Y 2 + Z 2 . Note the relation 1 2 2 M
M  MY2 X + MXZ + MZY = D2 4 that holds only on the solution spa e of (5). Every separable or R-separable solution set for (5), as well as the orresponding separable oordinates, is hara terized by a pair of se ond-order

ommuting onformal symmetry operators for (5). For ellipsoidal oordinates (R = 0) the operators an be hosen as 2 M
M + (a 1)PY2 + aPZ2 ; MXZ + aMY2 Z

whereas for oni al oordinates (R = 0) the operators are 2 M
M; MXZ + aMY2 Z :

12

aPZ2 ;

THE MODEL: We represent solutions (X; Y; Z ) of the Lapla e equation in the integral form (X; Y; Z ) =

Z

C1

d

Z

C~2

d' h[ ; '℄ exp [ (iX os ' + iY sin ' Z )℄

 I (h): The a tion of the s ale Eu lidean symmetries on the solution spa e of the Lapla e equation orresponds to the model operators

PX = i w ; PY = i w ; PZ = ; D =  + 21 ; MXY 1

2

= w2 w1 ;

MZX = iw  + iw w1 ; MZY = iw  iw w w1 : 2 2

1

2

1

2

(6)

where w1 = os '; w2 = sin ', so w12 + w22 = 1. Integral representation for solutions of the Lapla e equation that are eigenfun tions of the dilation operator D with eigenvalue ` 21 : 1 ) : 2 Choose C1 and C2 as unit ir les in the and t omplex planes, respe tively. Setting Dh = ( ` 21 )h we nd

D = ( `

h( ; t) =

`

1

j (t); j (t) =

`

X

m= `

am tm ; t = ei :

Then we evaluate the integral by residues to obtain (X; Y; Z ) = I (h) =



Z 2 0

[X os ' + Y sin ' + iZ ℄` j (ei' )d':

For j (t) = tm , `  m  ` we have M 0 = m so must be a multiple of the solid harmoni R` Y`m (; ), expressed in spheri al oordinates.

13

8 Non-Hilbert spa e expansion formulas Eigenvalue equations for the separable paraboli system of the Lapla e equation. In the two-variable model these eigenfun tions are elementary 1 f;m ( ; t) = e



2 tm

=

X

k

( =2)k k!

k

1

tm :

(7)

Choosing the ontours C1 ; C2 and mapping ba k to the solution spa e of the Lapla e equation, we nd ;m (X; Y; Z ) = 2

Z 2



0

 q

J0 i 2(Z

p



iX os  iY sin  eim d

p

= 4 2 Jm ( i  )Jm (  )eim ; where X =  os , Y =  sin , Z = ( 2  2 )=2. The fa t that the variables must separate in paraboli oordinates enables us to ompute the integral almost immediately. Furthermore, applying the transformation I su

essively to ea h term in the series (7), we nd the expansion ;m (X; Y; Z ) =

1

16 3im (=2)` Y`m (; ) `=jmj `! 4 (2` + 1)(` m)!(` + m)! X

q

of produ ts of Bessel fun tions in terms of spheri al harmoni s.

14

9 Models in higher dimensions Extend this analysis to Lapla e equations in n dimensions. For n = 4: (X2 + Y2 + Z2 + T2 ) = 0: (8) The onformal symmetry algebra of this equation is 15-dimensional, but for simpli ity we onsider only the 11-dimensional s ale Eu lidean subalgebra with basis

PX = X ; PY = Y ; PZ = Z ; PT = T MY X = MXY = Y X XY ; MXZ = ZZ ZX ; MZY = ZY Y Z ; MT X = T X XT ; MT Y = T Y Y T ; MT Z = T Z ZT ; D = (XX +Y Y +ZZ +T T ): Note the relation 2 2 M
M  MY2 X + MXZ + MZY + MT2 X + MT2 Y + MT2 Z = D(D

2)

holding only on the solution spa e of (8). The various spheri al and ellipsoidal separable solution sets for (8), are ea h hara terized by a triplet of se ond-order ommuting s ale Eu lidean symmetry operators for (8). Computations involving separable solutions of the Lapla e equation are simpli ed by making use of a 3 variable model for the solution spa e. Represent solutions (X; Y; Z; T ) in the integral form (X; Y; Z; T ) =

Z

C1

d

Z Z

D

exp [ (w1 X + w2 Y + w3 Z + iT )℄h( ; w)

 I (h); where w12 + w22 + w32 = 1.

15

dw1 dw2 w3 (9)

Integrating by parts, we nd that the a tion of the symmetries PX ;


; D on the solution spa e of the Lapla e equation orresponds to the operators

PX = w ; PY = w ; PZ = w ; PT = i ; D =  MXY = w w2 w w1 ; MXZ = w w1 ; MY Z = w w2 ; MT X = iw + iw  + i(1 w )w1 iw w w2 ; MT Y = iw + iw  iw w w1 + i(1 w )w2 ; MT Z = iw + iw  iw w w1 iw w w2 : 1

2

1

3

2

3

3

1

1

2

2

3

2 1

2

3

1

2

2 2

1

3

(10)

1

3

2

Integral representation for solutions of the Lapla e equation that are eigenfun tions of the dilation operator D with eigenvalue `: D = ` . Pro eeding as in the n = 3 ase, we nd (X; Y; Z; T ) = I (h) =

Z Z

D

[w1 X + w2 Y + w3 Z + iT ℄` j (w)

16

dw1 dw2 : w3 (11)

10 Niven operators Niven onstru ted an operator that maps harmoni fun tions into ellipsoidal solutions. Indeed, it maps a oni al oordinate solution to an ellipsoidal solution. A detailed te hni al proof is given by Whittaker and Watson. Here we give a mu h simpler proof, from the viewpoint of Lie theory. We treat an example of the ase n = 3 for simpli ity. Let H` be the spa e of solutions of the Lapla e equation that are homogeneous of degree `. There is an operator F` , the Niven operator, su h that the identities 



M
M + (a 1)PY2 + aPZ2 F` = F` (M
M)   2 2 MXZ + aMY2 Z + aPZ2 F` = F` (MXZ + aMY2 Z )

(12)

hold on H` . The operator an be hosen in the form

F` = 0 F1

`

1 2

!

1 ; ((a 4

1)PY2 + aPZ2 ) :

Thus the Niven operator is an intertwining operator on H`: If  satis es 2 the oni al eigenvalue equations (MXZ + aMY2 Z ) = 1  and M
M = `(` + 1), then = F`  satis es the ellipsoidal eigenvalue equations 



2 MXZ + aMY2 Z + aPZ2 = 1 ;





M
M + (a 1)PY2 + aPZ2 = `(` + 1) :

(13) (14)

We verify (12) using the model, on the spa e H` of fun tions h( ; t) = ` 1 j (t). Setting t = ei' we have (a 1)PY2 + aPZ2 = 2 (a sin2 ' + os2 '). Set F` = F` (x), where x = 2 (a sin2 ' + os2 '). Thus on H` the Niven operator is just multipli ation by an ordinary analyti fun tion of x. The rst equation (12) on H` then redu es to a se ond-order ODE for F` : 4xF`00 + ( 4` + 2)F`0

17

F` = 0:

(15)

The solution bounded at 0 is !

F` = F 0

`

1

1 2

1 ; x : 4

Transferring this operator over to the solution spa e of the Lapla e equation via F` = I (F`h) we obtain the required result. A lassi al result of Whittaker and Watson is that if Lm e ` () are Lam polynomials then

Lm` ()Lm` ( )Lm` ( )

=

K

Z 2

K

2

P` ()Lm` (Æ )dÆ

where

 = k2 sn sn sn snÆ (k2 =k02 ) n n n nÆ (1=k02 )dn dn dn dnÆ and P` (z ) is a Legendre polynomial. This result is a dire t onsequen e of the fa t proved above that if

H`m (X; Y; Z )

=

Z

 

(X os t + Y sin t + iZ )` f (t)dt;

then G` (X; Y; Z ) is also a solution of the Lapla e equation, where (transforming the a tion of the Niven operator from the model to the solution spa e of the Lapla e equation)

Gm` (X; Y; Z ) = ( 2

= 641 +

1` 2

1 2

D 1 `)( )`+ 2 I 2 1)r D2r

m ` 21 (D )H` (X; Y; Z )

( r=1 2
4


2r
(2` 1)
(2` 3)


(2` X

and D2 = Y2 + aZ2 .

18

3

2r + 1)

7 5

H`m (X; Y; Z )

11

Lame polynomials.

If (X; Y; Z ) satis es Lapla e's equation and is homogeneous of degree ` (an integer). Choose sphero oni al oordinates

i ik X = Rk sn(; k)sn(; k); Y = R 0 n(; k) n(; k); Z = R 0 dn(; k)dn(; k): k k (16) If we set = R` U ()V ( ) and substitute into the Lapla e equation, we obtain a solution if U () satis es the equation (2

k2 sn2 (; k) + )U^ () = 0;

with a similar result for V ( ). The operator whi h hara terizes  is =

1 2 2 M + MXZ : k2 XY

We now look for eigenfun tions of the same operator L = k12 M2XY + M2XZ in the two variable model. In this representation they must satisfy 

(

1 k2

sin2 ')'2

(2` + 3) sin ' os ''



(` + 1)(` + 2) os2 t + ` + 1  f (') = 0;

whi h an be transformed to the Lame equation. Mapping from the two variable model to the solution spa e of the Lapla e equation we obtain the produ t formula Z

L(; k)L(; k) = [ ikk0 sn(; k)sn(; k)sn(!; k)

k

n(; k) n(; k) n(!; k)+ k0

i dn(; k)dn(; k)dn(!; k)℄`L(!; k)d!; k where L(z; k) is a solution of the Lame equation (z2

k2 `(` + 1)sn2 (z; k) )L(z; k) = 0:

19

12 A general Heun type produ t formula. Consider the Lapla e equation in p+q +r +3 variables. We hoose oordinates of the form v u

s

u (u uv (u 1)(v 1) a)(v a) sp ; Y = R sq ; Z = Rt s X=R a 1 a a(a 1) r (17) where the sp an be expressed in terms of oordinates on a p dimensional sphere. We look for solutions of the p + q + r +3 dimensional Lapla e equation in these oordinates, i.e., r

(X2 + Y2 + Z2 ) = 0: If we take out the angular dependen e on the spheres and write (R; u; v ) = Rt U (u)V (v ) then variables separate. Using the two variable model we obtain a general produ t formula (sn(; k)sn(; k))

p=2



( n(; k) n(; k)) q=2 (dn(; k)dn(; k)) r=2 L()L( )P` (s1 )Pm (s2 )Pn (s3 ) Z Z Z k 1 = [ 0 dn(; k)dn(; k)dn(!; k)s3
s03 0 n(; k) n(; k) n(!; k)s2
s02 k2 s1 s2 s3 k 2 iksn(; k)sn(; k)sn(!; k)s1
s01 ℄` sn(!; k)p=2 n(!; k)q=2dn(!; k)r=2P` (s01 )Pm (s02 )Pn (s03 )ds01 ds02 ds03 d!: Here, the P` are spheri al harmoni s and the L are Heun fun tions.

20

13 Se ond-order superintegrability and multiseparability The maximal possible number of linearly independent 2nd-order symmetry operators for a S hrodinger equation H = E , where H =
n + V , is 2n 1. This maximum is a hieved only for a few remarkable manifolds and potentials.

De nition 2 The system H = E is se ond-order superintegrable if there are 2n 1 linearly independent se ond-order partial dierential symmetry operators: S` ; ` = 0; 1;


; 2n 2; S0 = H; [S` ; H ℄ = 0. Let's ompare the on epts of se ond-order superintegrability and separability on an n-dimensional Riemannian manifold. Superintegrable: There are 2n 1 linearly independent 2nd order onstants of the motion S0 = H; S1 ; S2 ;


; S2n 2 : [H; Sj ℄ = 0; j = 1;


; 2n 2. Separable: There are n linearly independent 2nd order symmetries S0 = H; S1 ; S2 ;


; Sn 1 ;: [Sj ; Sk ℄ = 0; 0  j; k  n 1. The symmetries must also satisfy eigenform onditions. One of the most ee tive methods of nding superintegrable systems is to sear h for systems that are multiseparable. Example : Real Eu lidean 2-spa e. Here, n = 2, 2n 1 = 3, so ea h separable system yields one new symmetry. Consider the S hrodinger equation with potential

k2 1 V (x; y ) = ! 2 (x2 + y 2 ) + 1 2 2 x

1 4

+

k22

y2

1 4

!

;

i.e., !

2 2 + x2 y 2

k2 ! (x + y ) + 1 2 x 2

2

2

21

1 4

+

k22

y2

1 4

!

= 2E :

This equation separates in three systems: Cartesian oordinates (x; y ); polar oordinates x = r os ; y = r sin , and ellipti al oordinates

x2 = 2

(u1

e1 )(u2 e1 ) ; (e1 e2 )

y 2 = 2

(u1

e2 )(u2 e2 ) : (e2 e1 )

The bound state energies are given by En = ! (2n + 2 + k1 + k2 ) for integer n. The orresponding wave fun tions are 1) Cartesian: 1 n1 ;n2 (x; y ) = 2! 2 (k1 +k2 +2) 2

!

s

1 1 n1 !n2 ! x(k1 + 2 ) y (k2+ 2 ) (n1 + k1 + 1) (n2 + k2 + 1)

2

e 2 (x +y ) Lkn11 (!x2 )Lkn22 (!y 2); and the Lkn (x) are Laguerre polynomials. 2) polar: 1 (r; ) = q(k1 ;k2 ) ()! 2 (2q+k1 +k2 +1)

e(

s

n = n1 + n2 ;

2m! (m + 2q + k1 + k2 + 1)

!r2 =2) r (2q+k1 +k2 +1) L2q+k1 +k2 +1 (!r 2 ); m v u u t

q(k1 ;k2 ) () = 2(2q + k1 + k2 + 1)

( os )k1

n = m + q;

q ! (k1 + k2 + q + 1) (k2 + q + 1) (k1 + q + 1)

(sin )k2 +(1=2) Pq(k1 ;k2 ) ( os 2); and the Pq(k1 ;k2 ) ( os 2) are Ja obi polynomials . 3) ellipti al: =e where

=

+(1 2)

n !(x2 +y2 ) xk1 + 21 y k2 + 21 Y

m=1

x2

y2

m

x2

e1

+

m

y2

!

e2

2

(u1 )(u2 ) :  e1  e2 ( e1 )( e2 ) A basis for the se ond order symmetry operators is

L1 = x2 +

( 14

+

k12 )

x2

2 = 2

! 2 x2 ;

L2 = y2 + 22

( 41

y2

k22 )

!2y2

1 y 2 1 2 x2 1 k12 ) 2 + ( k) : 4 x 4 2 y2 2 (Note that H = L1 + L2 .) The separable solutions are eigenfun tions of the symmetry operators L1 ; M 2 and M 2 + e2 L1 + e1 L2 . The algebra onstru ted by repeated ommutators is

M 2 = (xy

yx )2 + (

[L1 ; M 2 ℄ = [M 2 ; L2 ℄  R;

[Li ; R℄ = 4fLi ; Lj g + 16! 2 M 2 ;

i 6= j;

[M 2 ; R℄ = 4fL1 ; M 2 g 4fL2 ; M 2 g + 8(1 k22 )L1 8(1 k12 )L2 ; 64 8 R2 = fM 2 ; L1 ; L2 g + fL1 ; L2 g + 16! 2M 4 16(1 k22 )L21 3 3 128 2 2 16(1 k12 )L22 ! M 64! 2(1 k12 )(1 k22 ): 3 Note that these relations are quadrati . Here fA; B g = AB + BA is a symmetrizer. The important fa t to observe about the algebra generated by L1 ; L2 ; M 2 ; R is that it is losed under ommutation. One an use the stru ture of su h losed quadrati algebras to

ompute the spe tra of L1 ; L2 ; M 2 ; R algebrai ally! In real Eu lidean two-spa e there are pre isely four potentials that have the multiseparation property. The se ond potential is ( 41

V (x; y ) = ! 2 (4x2 + y 2)

k22 )

y2

:

The orresponding S hrodinger equation is separable in Cartesian oordinates and paraboli oordinates x = 12 ( 2  2 ); y = : The third potential is

 B V (x; y ) = p 2 2 + 1 4 x +y

q

px + y + x px + y x B px + y + 4 px + y ; q

2

2

2

2

2

2

2

2

2

separable in paraboli and paraboli oordinates of the se ond type x = ; y = 12 (2  2 ): The fourth potential is

 1 V (x; y ) = p 2 2 + p 2 2 x +y 4 x +y

separable in polar, paraboli and modified ellipti oordinates. 23

!

1 1 2 2 px(2k1+ y24+) x + px(2k2+ y24 ) x ;

For n = 2, a omplete lassi ation of all se ond-order superintegrable potentials ( lassi al and quantum) has been given for real and omplex Eu lidean spa e and real and omplex spheres. All these ases share the same basi features: 1. The potential V permits separability of the S hrodinger equation H = E in at least two oordinate systems, hara terized by symmetry onditions L1 = 1 , L2 = 2 . Superintegrability implies multiseparability. (2) 2. One an obtain alternate spe tral resolutions f (1) j g; f k g for the multiply(1) (1) (2) (2) (2) degenerate eigenspa es of H , L1 (1) j = 1 j , L2 k = 2 k . These alternate resolutions resolve the bound state degenera y problem. P (1) 3. The interbasis expansions (2) k = j ajk j yield important spe ial fun tion identities. 4. The operators H; L1 ; L2 generate a quadrati algebra. With R = [L1 ; L2 ℄, R2 is a polynomial of order 3 in H; L1 ; L2 , whereas [L1 ; R℄ and [L2 ; R℄ are of order 2 in H; L1 ; L2 . Closure of these algebras is a remarkable property, and is false for general symmetries. The quadrati algebra stru ture an be used to ompute the interbase expansion oeÆ ients.

24

14 Example of superintegrabiltiy in n dimensions: A \magi " potential on the n-sphere Introdu e Cartesian oordinates z0 ; z1 ;


; zn in n + 1 dimensional Eu lidean spa e and restri t these oordinates by the onditions

z02 = 1

n

X

i=1

xi  xn+1 ; z12 = x1 ; z22 = x2 ;




; zn = xn: 2

Note that z + z +


+ zn2 = 1. De ne a metri by ds2 = nm=0 (dzm )2 . The spa e orresponds to a portion of the n-sphere S n. Consider the Lapla eBeltrami equation MODEL ONE : (
n + Vn (x)) = M (M + G 1) ; where " # +1 ( i 21 )( i 32 ) 1 1 nX (n 3)(n + 1) 2 Vn = : + (1 G) 1 4 i=1 xi 4 4 P +1 and the i are real onstants with G = ni=1

i and M = 0; 1;


. This equation has a natural metri 1=2 d! = g 1=2 dx1


dxn = x1 1=2


xn 1=2 xn+1 dx1


dxn : Set H =
n + n and n X 1 n+1  1 = x 11 =2 1=4


xn n =2 1=4 x nn+1+1 =2 1=4 ; n = [ j +( G)xj ℄xj : 2 2 j =1 Then with (x) = (x) (x) we see that MODEL TWO : (
n + n ) = M (M + G 1) () (
n + Vn(x)) = M (M + G 1) : Furthermore the eigenspa e of H orresponding to eigenvalue M (M + G 1)

onsists entirely of polynomials in the xj . This indu es an inner produ t on the spa e of polynomial fun tions (x) =  , with respe t to whi h H is self-adjoint: 2 0

P

2 1

(1 ; 2 ) < 1 ; 2 >=

Z






Z

xi >0

1 2 d!~ ;

d!~ = x 11 1 : : : x nn 1 x nn+1+1 1 dx1 : : : dxn ; (H 1 ; 2 ) = (1 ; H 2 ): 25

REMARKS: 1. If 1 = 2 =


= n+1 = 1=2 then H =
n and that the Lie algebra of real symmetry operators of
n is so(n + 1), with dimension n(n + 1)=2 and a basis of the form fL`k g where 0  ` < k  n, and L`k = Lk` . Expli itly, L`k = z` zk zk z` ; and Lij = 2pxipxj (xj xi ); 1  i; j  n L0i = 2 xi xn+1 xi ; 1  i  n: 2. For 1 ; : : : ; n+1 arbitrary, the only rst order symmetry is multipli ation by a onstant. However, there are se ond order symmetries (Chesire at ee t):

Sij  4xi xj (xi

xj )2 + 4( ixj

j xi )(xi

xj )

1 1 )xj ( j )x ℄(  ) = Sji ; 1  i < j  n; 2 2 i xi xj S0i  4xi xn+1 x2i + 4[ ixn+1 n+1 xi ℄xi 1 1 = L20i + 4[( i )xn+1 ( n+1 )x ℄ = Si0 ; 1  i  n: 2 2 i xi

= L2ij + 4[( i

26

14.1

Orthogonal bases of separable solutions

All separable oordinates on the n-sphere are known, i.e., all separable oordinates for the Lapla e-Beltrami eigenvalue equation
n =  . They an be onstru ted by a graphi al pro edure. We know that:



For n = 2 there are two separable systems (ellipsoidal and spheri al

oordinates), while for n = 3 there are 6 systems. The number of separable systems grows rapidly with n, but all systems an be onstru ted through a simple graphi al pro edure.



The equation (
n + Vn) =  where the s alar potential takes the spe ial form

Vn =

n

i 0 + 2 ; 0 ; 1 ; : : : ; n onstants:; 2 z0 i=1 zi

X

is separable in all the oordinate systems in whi h the Lapla e-Beltrami eigenvalue equation is separable. Indeed, the equation with this potential is separable in general ellipsoidal oordinates and their limits. EXAMPLE: Spheri al oordinates fui g on S n

z02 = z12 = z22 = .. . zn2 1 = zn2 =

1 x = 1 un x1 = u1 u2 : : : un x2 = (1 u1 )u2 : : : un

xn 1 = (1 un 2 )un 1un xn = (1 un 1)un :

(Note that in terms of angles fi g one usually sets ui = sin2 i .)

27

14.2

Relations between bases on the sphere

Take the ase n = 2 in MODEL TWO. Then

H  = j (j + G 1); where

H=

2 X

(xi Æik

xi xk )

i;k=1

2 X 2 + ( i xi xk i=1

Gxi )

 : xi

Here G = 1 + 2 + 3 . FACTS:

H

maps polynomials of maximum degree mi in xi to polynomials of the same type.



The polynomial eigenfun tions of H form a basis for the spa e of all polynomials f (x1 ; x2 ).



The spe trum of H a ting on this spa e is exa tly f j (j + G 1) : j = 0; 1; : : :g.

 H is self-adjoint with respe t to the inner produ t (f1 ; f2 ) =

Z






Z

x1 ;x2 >0;1 x1 x2 >0

f1 (x)f2 (x)d!

where

dw = x 11 1 x 22 1 (1 x1

x2 ) 3 1 dx1 dx2 ; (Hf1 ; f2 ) = (f1 ; Hf2 );

and 1 ; 2; 3 are positive real numbers. Here f1 ; f2 are polynomials in x = (x1 ; x2 ).



There are exa tly two separable oordinate systems for this equation: spheri al oordinates and ellipsoidal oordinates.

Spheri al oordinates: For xed j the polynomials jm (x1 ; x2 )

= (x1 + x2 )m Pj 1 +m 2 +2m Pm 2 1; 1 1 x12+x1x2 28

; 3

1

(2x1 + 2x2 1) 1 ; m = 0; 1; : : : ; j 1



form an orthogonal basis for the eigenspa e orresponding to eigenvalue j (j + G 1). Note that variables separate in spheri al oordinates. Ellipsoidal oordinates: For ellipsoidal oordinates fx; y g we have

zi2 =

(x ei )(y ei ) ; i = 1; 2; 3; i; j; k pairwise distin t: (ej ei )(ek ei )

The separation equations are h

2



[ ( e1)( i e2 )( e3 ) dd 2 +  1e1 +  2e2 +  3e3 dd + j (j + G 1) + q ℄jq () = 0 where  = x; y a

ording as  = 1; 2; respe tively. This is Heun's equation, the Fu hsian equation of se ond order with four singularities. The solutions for the fun tions jq () are Heun polynomials whi h for xed j will form a omplete set of basis fun tions on e the eigenvalues q have been al ulated. To al ulate the eigenvalues it is onvenient to observe that in the oordinate system x1 ; x2 the operator M whose eigenvalues u are

u = 4q

(e1 + e2 + e3 )j (j + G

1)

is given by

M = (e

1

+ e2 )S12 + (e2 + e3 )S23 + (e1 + e3 )S13

where the Sik are the symmetry operators. That is, M is the se ond order symmetry operator for the Lapla ian ([M;
℄ = 0) whi h orresponds to the separable oordinates x; y , and the Heun basis = 1jq (x)2jq (y ) is

hara terized as the set of eigenfun tions M = u .

29

Expanding the Heun basis 1jq (x)2jq (y ) in terms of the Ja obi polynomial basis: j X 1 2 m jm [; ℄: = jq (x)jq (y ) = m=0

Three term re urren e relations for the expansion oeÆ ients m an be dedu ed by requiring that M =u : To arry out the omputation we need the a tion of the various pie es Sik of M on the Ja obi bases jm [; ℄. We nd

M

jm [; ℄

=

+1 X

r=

Xr

j;m+r [; ℄;

1

e2 )( 1 + 2 + 3 + m + j 1)( 3 m + j 1)(m + 1) ( 1 + 2 +m 1); ( 1 + 2 + 2m 1)( 1 + 2 + 2m) with similar expressions for X 1 (m; j ); X0 (m; j ). Substituting this expansion into the eigenvalue equation M = u we nd the three term re urren e relation

X1 (m; j ) =

4(e1

X1 (m 1; j )m 1 + (X0 (m; j ) u) m + X 1 (m + 1; j )m+1 = 0 where m = 0; 1; : : : j . Consequently the j + 1 independent eigenvalues q are

al ulated from the determinant

X0 (j; j ) u X1 (j 1; j ) X 1 (j; j ) X0 (j 1; j ) u ... ...

X1 (j

...

2; j )

X 1 (1; j ) X0 (0; j ) u = 0:

30



MAIN ISSUES: 1. The importan e of models. 2. Classi ation of superintegrable systems. 3. The relationship between superintegrabiltiy and multiseparability. 4. The stru ture of quadrati algebras.

31