Multiseparability and Superintegrability for Classical and Quantum Systems

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

Multiseparability and Superintegrability for Classical and Quantum SystemsInstitute for Mathematics and its Applications University of Minnesota, Minneapolis, Minnesota, 55455, U.S.A.February 17, 2000 W. Miller, Jr.

Abstract It has long been known that there are potentials on n-dimensionalconstant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in more than one coordinate system. Smorodinsky, Winternitz et.al., initiated the methodical search for such potentials in two and three dimensions, and there has been a considerable amount of work for various examples. Such a system is called maximal in dimension n if there exist 2n? 1 functionally independent integrals of motion. In some papers, these systems are called superintegrable. In the rst part of this paper we outline the basic ideas relating to the notion of superintegrable potentials. The energy observable is degenerate for potentials of this type and the corresponding intergrals of motion that arise from the simultaneous separability close quadratically under repeated commutation. We give examples of these systems and indicate how superintegrability can be of use, particularly in relation to bound states. Virtually all of the special functions of mathematical physics (in one and several variables) arise in this study and formulas expanding one type of special function as a series in another type emerge as a byproduct. Finally, we describe how one can, in principle, classify all such systems and deduce the structure of the quadratic algebra. Many of the results reported 1

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

here were obtained in collaboration with E.G. Kalnins and G.S. Pogosyan.

1 IntroductionIt has long been known that Schrodinger's equation with certain special potentials can admit (multiplicative) separation of variables in more than one coordinate system. This is intimately related to the notion of superintegrability, 1, 2, 3]. This subject has been studied by a number of authors, based on the use of the corresponding di erential equations that that are implied by the requirement of simultaneous separability, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Speci cally, superintegrability means that for a Schrodinger equation in dimension n there exist 2n? 1 functionally independent quantum mechanical second-order observables (i.e., second-order self-adjoint operators that commute with the Hamiltonian). There is an analogous concept of superintegrability for classical mechanical systems. This relates to the corresponding additive separation of variables of the Hamilton-Jacobi equation. Furthermore, one observes that if we do have simultaneous separability then the resulting constants of the motion close quadratically under repeated application of the Poisson bracket, 11]. We also know that for spaces of constant curvature separable coordinate systems of the free motion are described by quadratic eleme

nts of the corresponding rst order symmetries, 16, 17, 18]. Although concrete examples of superintegrable systems are easily at hand, a complete classi cation of all such systems has presented major di culties. How can one be sure that all systems for free motion have been found? (For example, Ra~ada's classi cation 15] omits our system 5 below.) Once these n are determined, how can one be sure that the most general additive potential term has been calculated? Here we will present the background information to understand the problem, and its importance, and present a new approach to its solution, with details for two dimensional complex Euclidean space. Consider an n-dimensional Riemannian manifold Rn. (In most of the following we will assume that Rn is a space of constant curvature, for that is the case where the most interesting and rich applications arise.) In local coordinates q;; qn the contravariant metric tensor is gjk(q) . Let V (q) be a potential function on Rn. The corresponding Hamilton-Jacobi equation1

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

is wheren X j;k=1

H(q; p)= E H(q; p)=gjk(q)pj pk+ V (q)=n X j;k=1

(1)

@S@S gjk(q)@q@q+ V (q);j k

(2)

and S (q) is the action function, 19]. The quantum analog of this classical system is given by the Schrodinger equation H (q)= E (q) (3) where in local coordinates n 1 X@ (pggjk )@ H= n+ V (q); (4) n=p g j;k@qj@qk=1

and g= det(gjk)? . Recall that a complete integral S (q;;; n) of the Hamilton-Jacobi equation solves the associated classical mechanical system 19, 20]. (A complete integral is a solution of (1) such that locally1 1

!@ S 6= 0: det@qj@ k2

Any solution of the Hamilton-Jacobi equation via (additive) separation of variables n X S (q;;; n)= S j (qj; )1 ( )

where= E;;; n are the separation constants, yields a complete integral. Similarly, in the quantum case, if the Schrodinger equation H= E (multiplicatively) separates in the coordinates q then we can write1 2

j=1

(q)=

n j=1

( )

j

(qj; )

(5)

and this ansatz allows the decomposition of (3) into n ordinary di erential equations, one for each of the factors j . Many of the special functions of( )

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

mathematical physics occur as solutions of these ordinary di erential equations. For orthogonal coordnates q on an n-dimensional constant curvature space (i.e., such that gjk= 0 for j 6= k) one can show that the HamiltonJacobi equation is additively separable if and only if the Shrodinger equation is multiplicatively separable. (See 21] for a discussion of the relationship for general Riemannian manifolds.) We shall see that superintegrability is closely linked to symmetry properties of (1) and (3), and to separation of variables (special function) solutions of (3). At this point it is useful to summarize brie y the history of the symmetry/special function approach to solving the Schrodinger equation (3). (Superintegrability is just one of the latest chapters.) Beginning with the introduction of the Schrodinger equatio

n in the 1930's, and continuing until the 1960's, the main emphasis was on the study of Lie symmetry groups of unitary operators that commuted with the Hamiltonian H, hence mapped solutions of (3) to solutions. At the Lie algebra level, one looked for algebras of rst-order di erential operators L that commuted with H: L; H] LH? HL= 0. (Again L maps solutions of (3) to solutions.) This led to studies of rotationally invariant potentials and the theory of spherical harmonics 22]. A related concept was that of dynamical symmetry groups or Lie algebras. The idea was to imbed H as an element of a Lie algebra of rst and second-order di erential operators. The representation theory of the Lie algebra could then be used to derive information about the eigenvalues and eigenvectors of H . The harmonic oscillator and the Morse potential were treated in this way. The so-called factorization method for solving the Schrodinger equation is related to this approach 23]. Among the special functions that arise and whose properties can be studied from this connection are Bessel functions and (more generally) hypergeometric functions. (More recently, q-analogs of the dynamical symmetry algebra approach have led to q-hypergeometric functions 24].) Since the 1960's we have been in the\Cheshire Cat" era, 25]. In the most recent theories relating integrabilty, superintegrability and variable separation, the Lie groups and algebras have disappeared, but their grin persists,^. The focus here is on second order constants of the motion (symmetry operators that are built out of products of rst order Lie symmetries for the zero potential problem) and their connection with variable separation for 4

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

both the Hamilton-Jacobi and Schrodinger equations. One of the seminal papers in this regard was by Smorodinsky et al. 5]. Some other contributions are, for example, by Shapovalov, Kalnins and Miller, and Winternitz 26, 27, 28].These approaches also exploit the maximum symmetry of the physical system, but no longer in terms of Lie algebras of operators. To examine these ideas, let us start with a classical system X H= gjk pj pk+ V (q) (6) where the pj are the momenta conjugate to the coordintes qj . Recall that the Poisson bracket of two functions fh(q; p), h= 1; 2 is the function n X ff; f g(q; p)= (@f@f?@f@f ); (7) j@qj@pj@pj@qj 20]. A second-order constant of the motion for (6) is a function X L= ajk (q)pj pk+ W (q); ajk= akj; (8) such that fL; Hg= 0. Note that the null space of the map T: df (q; p) ! ff; Hg(q; p) is 2n?1 dimensional. Thus (locally) there are 2n?1 functionally independent constants of the motion (but not necessarily second-order). For the purposes of this paper we adopt the following de nitions. We say that the classical system H= E is superintegrable or maximal if there are 2n? 1 functionally independent second-order constants of the motion: X L`= ajk (q)pj pk+ W`(q) L= H;`= 0; 1;; 2n? 2 fL`; Hg= 0: (9) We say that the quan

tum system H= E is superintegrable or maximal if there are 2n? 1 linearly independent second-order symmetry operators: X 1 p jk L`= pg@qj ( ga (q)@qk+ W`(q) L= H;`= 0; 1;; 2n? 2 L`; H] L`H? HL`= 0: (10)1 2 1 2 1 2=1 0 0

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

To clarify the connection between these ideas and variable separation we assume that the coordinates q are orthogonal, i.e., the covariant metric tensor is diagonal:

X X ds= gjk dqj dqk= Hj (q) dqj; so that the Hamilton-Jacobi equation is given by X@S H= Hj? (@q )+ V (q)= E j j2 2 2 2 2

(11)

@S Set@qj= Sj= pj and assume additive separation in the q coordinates, so that@j Si=@j@i S= 0 for i 6= j . The separation equations are postulated to be

Si?2

n X

j=1

uij (qi) j+ fi (qi)= 0;

i= 1;

; n;

1

= E:

(12)

Here@k uij (qi)= 0 for k 6= i and det(uij ) 6= 0. We say that U= (uij ) is a Stackel matrix. Then (11) can be recovered from (12) provided Hj?= (U? ) j . The quadratic forms2 1 1

L`=satisfy

n X

j=1

(U?1 )`j (p2+ fj

j (qj ))=

n X

j=1

(U? )`j pj+ W`(q)1 2 2 2

Hj pj+ V (q) for a separable solution. Furthermore, we have fL`; Lj g= 0;` 6= j Thus the L`; 2` n, are constants of the motion for the Hamiltonian L. An analogous construction, replacing (12) by n second-order linear ODE's for factors i (qi) leads to second order linear partial di erential operators L= H; L;; Ln such that H= E; L`=`;`= 2;; n (13)

L`=?`;

H=L=1

X

1

( )

1

2

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

and

Then one can verify that Lk; H]= 0; Lk; Lj]= 0: How does one nd all orthogonal separable coordinate systems q for a given space Rn for zero potential, V 0? This is a di cult problem in di erential geometry. The answer is known for some constant curvature spaces. In real Euclidean 2-space there are four separable systems: cartesian, polar, parabolic and elliptic. For complex Euclidean 2-space, including real Euclidean space and real Minkowski space, there are six 4, 16, 18]: Cartesian, polar, parabolic, elliptic, hyperbolic and semi-hyperbolic. We describe these coordinate systems and their corresponding free particle constants of the motion L. (We adopt the basis px; py; M= xpy? ypx for the Lie algebra e(2; C ) and de ne p= px ipy,^.) There is one orbit of constants of the motion, with representative Mp, that is not associated with variable separation 21]. The separable systems are:+

(q)=

n k=1

( )

i

(qi ):

Cartesian coordinates

Polar Coordinates

x; y;

L= px2

(14)2

Parabolic Coordinates. 1 xP= 2 (? ); yP=; L= Mpy Elliptic Coordinates (in algebraic form) xE= c (u? 1)(v? 1); yE=?c uv;2 2 2 2 2 2

x= r cos; y= r sin;

L=M

(15) (16) (17)

Hyperbolic Coordinates2 2

L= M+ c px2 2 2 2 2

s s xH= r+ r rs+ s; yH= i r? r rs+ s; 2 2 L= M+p2 2 2 2 2 2+

(18)

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

Semi-Hyperbolic Coordinates1 1 xSH=? 1 (w? u)+ 2 (w+ u); iySH=? 1 (w? u)? 2 (w+ u);(19) 4 4 L= 2Mp+ p?2 2+ 2

In real Euclidean 3-space there are 11 separable systems see T

able 1, 4, 16]. On the real 2-sphere there are 2: spherical and ellipsoidal. For real n-dimensional Euclidean space and the n-sphere Kalnins and the author have a graphical procedure to classify and construct all possibilities, 17, 18]. On the 2-hyperboloid there are 9 separable systems, 18]. For the n-hyperboloid of two sheets there is again a graphical procedure to construct all possibilites, 18]. In each case above, the symmetries Lj are second order elements in the enveloping algebra of the symmetry Lie algebra of the corresponding manifold, e.g., the Lie algebra e(n; C ) for Euclidean n-space and so(n+ 1; C ) for the n-sphere,^. We see that for zero potential, each of the constant curvature spaces listed above is separable in multiple coordinate systems. Indeed we can veryfy that the zero potential is superintegrable on each of these spaces. However, a potential V 6= 0\breaks the symmetry" and reduces the number of separable systems, usually to zero. (See 21] for conditions that must be satis ed by a potential in order to permit separation in a given coordinate system.) How does one determine which constants of the motion lead to variable separation?

Theorem 1 Necessary and su cient conditions for the existence of an orthogonal separable coordinate system fqi g for the Hamilton-Jacobi equation H= E on an n-dimensional kRiemannian manifold are that there exist n P quadratic forms Lk= n Lij pi pj+ Wk on the manifold such that: i;j 1. fLk; L`g= 0; 1 k; i n, 2. The set fLk g is linearly independent (as n quadratic forms). 3. There is a basis f! j: 1 j ng of simultaneous eigenforms for the n matrices fLijk g.1 ( )=1 ( ) ( )

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

Table 1 Separable coordinates in 3-D real Euclidean space.Coordinate System I. Cartesian x; y; z 2 R II. Cylindrical polar> 0, ' 2 0; 2 ) III. Cylindrical elliptic z 2 R, e<< e< IV. Cylindrical parabolic; x 2 R, 0 V. Spherical r> 0; 2 0;], ' 2 0; 2 ) VI. Prolate spheroidal e<u<e<u, ' 2 0; 2 ) VII. Oblate spheroidal e<u<e<u, ' 2 0; 2 ) VIII. Sphero-conical r 0 e<<e<<e IX. Parabolic; 0, ' 2 0; 2 ) X. Ellipsoidal a<u<a<u<a<u1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 3 1 1 2 2 3 3

Coordinates

x; y; z x= cos ', y= sin ', z x=2 (

1?e1 )( 2?e1 ), (e2?e1 )1 2

y=2 2 2

(

1?e2 )( 2?e2 ), (e1?e2 )

z

x, y=, z= (? ) x= r cos cos ', y= r sin sin ', z= r cos x= z=2 2

u1?e2 )(u2?e2 ) cos2 ', (e1?e2 ) (u1?e1 )(u2?e1 ) (e2?e1 )(

y=2

(

u1?e2 )(u2?e2 ) sin2 ', (e1?e2 ) u1?e1 )(u2?e1 ) sin2 ' (e2?e1 )

x= z=2 2 2 2

u1?e1 )(u2?e1 ) cos2 ', (e2?e1 ) (u1?e2 )(u2?e2 ) (e1?e2 )( 2( ( 2( (

y=2 2( (

(

x=r z=r x= x= z=2 2 2 2 (

1?e1 )( 2?e1 ) e1?e2 )(e1?e3 ), 1?e3 )( 2?e3 ) e3?e2 )(e3?e1 )

y=r2

1?e2 )( 2?e2 ) e2?e1 )(e2?e3 )

cos ', y=

sin ', z= (? )1 2 2 2

u1?a1 )(u2?a1 )(u3?a1 ), (a3?a1 )(a2?a1 ) (u1?a3 )(u2?a3 )(u3?a3 ) (a1?a3 )(a2?a3 )( )( ( )( ) ) 1 2

y=2 2

(

u1?a2 )(u2?a2 )(u3?a2 ) (a1?a2 )(a3?a2 )1

?a2 )( 2?a2 )( 3?a2 ) (a2?a3 )

XI. Paraboloidal 0<<a<<a<1 2 2 3

3

x= 1?a3 a2?a3 3?a3, y= 3?a2 z= (++?a?a )1 2 3 2 3

(

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

If conditions (1)-(3) are satis ed then there exist functions gi (q) such that:

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

; n:

Theorem 2 Necessary and su cient conditions for the existence of an orthogonal separable coordinate system fqi g for the Schrodinger equation ( n+di erential operators on the manifold such that: 1. Lk; L`]= 0; 1 k;` n, 2. Each Lk is in self-adjoint form, 3. There is a basis f!(j): 1 fLk g.1 2

V )= E on an n-dimensional constant curvature space are that there exists a linearly independent set fL= H= n+V; L;; Ln g of second-order

j

ng of simultaneous eigenforms for the

If conditions (1)-(3) are satis ed then there exist functions gi (q) such that:

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

; n:

See 28, 21] for proofs and discussions of these theorems. The main point of the theorems is that, under the required hypotheses the eigenforms !` of the quadratic forms Lij are normalizable, i.e., that up to multiplication by a nonzero function, !` is the di erential of a coordinate. This fact permits us to compute the coordinates directly from a knowledge of the symmetry operators. For general Riemannian manifolds Theorem 1 remains true, but Theorem 2 is false unless separation is replaced by the more general concept of R-separation 21]. We expect a superintegrable system to separate in multiple coordinate systems, though the above remarks do not constitute a proof of this. Thus, one way to nd superintegrable systems is to search for potentials V (q) that permit separation in multiple coordinate systems.

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

2 Examples for the Euclidean planeTo illustrate the basic ideas we can consider the example of the Schrodinger equation with potential ! 1 ! (x+ y )+ k?+ k?; V (x; y)= 2 x y i.e., ! ! k? k?@+@? ! (x+ y )+ x+ y=?2E:@x@y This equation separates in three coordinate systems: Cartesian coordinates (x; y); polar coordinates x= r cos; y= r sin, and elliptical coordinates )( )(? x= c (u?ee?u )? e ); y= c (u?ee?u ) e ): ( e ( e The bound state energies are given by En= !(2n+ 2+ k+ k ) for integer n. The wave functions for each of these coordinate systems are: 1. Cartesian coordinates s 1 k1 k2 1 n !n ! 2 x k1 2 y k2 1 n1;n2 (x; y )= 2! 2?(n+ k+ 1)?(n+ k+ 1) 2 e? ! x2 y2 Lk11 (!x )Lk22 (!y ) n n k (x) are Laguerre polynomials, 29] where n= n+ n, and the Ln 2. polar coordinates s 1 2m! k1;k2 ( )! 2 q k1 k2 (r; )= q?(m+ 2q+ k+ k+ 1) e?!r2= r q k1 k2 Lmq k1 k2 (!r ) where n= m+ q,2 2 2 2 1 1 4 2 2 1 4 2 2 2 2 2 2 2 2 1 1 4 2 2 1 4 2 2 2 2 2 2 1 1 2 1 2 2 1 2 2 2 1 2 2 1 1 2 (++2) 1 2 (+ ) (+ ) 1 1 2 2 (+ ) 2 2 1 2 ( ) (2+++1) 1 2 ( 2) (2+++1) 2+++1 2

v u k1;k2 ( )= u2(2q+ k+ k+ 1) q !?(k+ k+ q+ 1) t q?(k+ q+ 1)?(k+ q+ 1)( ) 1 2 1 2 2 1

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

(cos )k1= (sin )k2= Pq k1;k2 (cos 2 ); and the Pq k1;k2 (cos 2 ) are Jac

obi polynomials, 29]. 3. elliptical coordinates ! 1 1 n x+ y?c?! x2 y2 xk1 2 y k2 2 Y=e m?e m?e m where use has been made of the identity x+ y? c=?c (u? )(u? ):?e?e (? e )(? e ) The zeros j satisfy the relations k+1+ k+1+ X 2? != 0: m?e m?e j6 m ( m? j )+(1 2)+(1 2) ( ) ( ) (+ )++ 2 2 2=1 1 2 2 2 2 2 1 2 1 2 1 2 1 2 1 2=

(20)

Associated with the separability of the Schrodinger equation in these coordinate systems there are second order symmetry operators. A basis for such operators is (?k ) (?k ) L=@x+ x? ! x; L=@y+ y? ! y y 1 M= (x@y? y@x)+ ( 4? k ) x+ ( 1? k ) x? 1: 4 y 2 (Note that H= L+ L .) The separable eigenfunctions already given are eigenfunctions of the symmetry operators L; M and M+ e L+ e L with eigenvalues c=?! (2n+ k+ 1); p= (2q+ k+ k+ 1)+ (1+ k+ k ); e= 2(1? k )(1? k )? 2e ! (k+ 1)? 2e ! (k+ 1)? ! e e? q X k+1+1 4 e? e+ e k? e]: m m m The algebra constructed by repeated commutators is (R is de ned by the rst relation) L; M]= M; L]= R; Li; R]=?4fLi; Lj g+ 16! M; i 6= j;1 2 1 4 2 1 2 2 2 2 2 1 4 2 2 2 2 2 2 2 1 2 2 2 2 2 2 1 2 1 2 1 1 2 1 1 1 2 2 2 1 2 2 1 2 2 1 1 2 2 1 2 2 1=1 1 1 2 2 1 2 2

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

M; R]= 4fL; M g? 4fL; M g+ 8(1? k )L? 8(1? k )L; 8 R= 3 fM; L; L g+ 64 fL; L g+ 16! M? 16(1? k )L 3?16(1? k )L? 128 ! M? 64! (1? k )(1? k ): 3 These relations are quadratic. In real Euclidean two-space there are precisely four potentials that have the multiseparation property, 14]. The second potential is (?k ) V (x; y)= ! (4x+ y )? y: The corresponding Schrodinger equation is separable in two coordinate systems: Cartesian coordinates and parabolic coordinates 1 x= 2 (? ); y=: The third potential is ! (k? ) (k? ) 1 px+ y+ x+ px+ y? x:+ p V (x; y)=? p x+y 4 x+y The corresponding Schrodinger equation is separable in two coordinate systems: polar, parabolic and modified elliptic coordinates, where= c (u? e )(u? e ):= c (u? e )(u? e ); (e? e ) (e? e ) This last coordinate system can be written as v v u (U? E )(U? E ) q u (U? E )(U? E ) u u t; y= t 4(E? E )?2 E?E; x= 4(E? E )1 2 2 2 1 2 1 2 2 1 2 1 2 2 2 2 2 2 1 2 1 2 2 2 2 2 1 2 2 2 2 2 1 4 2 2 2 2 2 2 1 2 2 2 2 2 1 4 2 2 2 2 1 4 2 2 2 1 1 2 1 2 2 1 2 2 2 1 2 2 1 1 1 2 1 1 2 2 2 1 2 2 1 1 2

where E=?e e; E=? (e+ e ) and Uj= uj? uj (e+ e ): The fourth potential is1 1 2 2 1 4 1 2 2 2 1 2 1 2 2 2 2 2 2 2 2 2

qp qp x+y+x B p+y?x x V (x; y)=? px+ y+ B px+ y+ 4 4 x+y:2 2

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

Separation occurs here in parabolic and parabolic coordinates of the second type 1 x=; y= 2 (? ): As an illustration of the utility of the notion of a quadratic algebra consider the last potential given. A basis for the quadratic algebra consists of L; L and H with de ning relations 1 R; L]=?4L H+ B B; R; L]= 4L H+ 2 (B? B ) R= 4L H+ 4L H? 16 H+ (B? B )L? 2B B L? 2 (B+ B ) with R= L; L]. If we look for eigenfunctions of the

operators L; L respectively, we have L 'm= m 'm; L n= n n: If we write X L n= Cn2 2 1 2 1 2 1 2 2 1 2 1 2 2 2 2 1 2 2 2 2 2 2 1 1 1 2 2 2 2 1 2 2 1 2 1 2 1 2 1

Cn C (2? n? )= (8E n+ B B+ 16 E ) n: These relations in turn imply that (B? B )+ 16 E Cnn=? 8E and Cnn= Cn n are the only nonzero coe cents. Indeed they can essentially be determined by the relation p 4?2E (jCn;n j? jCn?;nj )= 8E n+ B B+ 16 E where the eigenvalues m and n are given by (B+ B )? (2n+ 1)p?2E B? (2m+ 1)p?2E; n=2? m=2? 8E 16E and the quantisation condition for E is p 4? B 8+ B=?(q+ 2)?2E E for integer q.1 2 1 2 2 1 2 2+1+1+1 2 1 2 1 2 2 1 1 2 2 2 1 2 2

then the quadratic algebra relations imply 1 ( n? )+ 8E]Cn=? 2 (B? B )? 16 E]2

X

2 1

2 2

n

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

3 More examples: The real 2-sphereThese ideas work also for separable coordinates on the real two dimensional sphere, 14]. An important example is The potential"# 1 k?+k?+k? V=2 s s s where s+ s+ s= 1. The corresponding Schrodinger equation has the form2 1 2 1 1 4 2 2 2 2 1 4 2 3 2 3 1 4

#@? s@ )+ (s@? s@ )+ (s@? s@ )+ (s@s@s@s@s@s@s"# (k? ) (k? ) (k? )=?2E: s+ s+ s This equation admits solution via separation of variables in two coordinate systems: spherical coordinates"1 2 2 2 1 1 3 3 2 1 3 2 2 2 3 2 1 1 4 2 2 1 4 2 3 1 4 2 1 2 2 2 3

2 1

2 2

2 3

s= sin cos '; s= sin sin '; s= cos1 2 3

and elliptical coordinates?e?e) si= (u? ei)(u? e i); (e j )(ei j2 1 2 1

i; j; k= 1; 2; 3

i 6= j 6= k:

Indeed a basis for the second-order symmetries of Schrodinger's equation with this potential is

s s 1 1 1 Lij= (si@sj? sj@si )+ ( 4? ki ) sj+ ( 4? kj ) si? 2; i 6= j: i j2 2 2 2 2 2 2

These symmetries satisfy the quadratic algebra relations

Lij; R]= 4fLij; Ljk g? 4fLij; Lik g+ 8(1? ki )Ljk? 8(1? kj ) Lik; 4 R=? 3 fLij; Lik; Ljk g+ 64 fLij; Lik g+ 64 fLij; Ljk g+ 64 fLik; Ljk g? 3 3 3 16(1? kk )Lij? 16(1? kj )Lik? 16(1? ki )Ljk+2 2 2 2 2 2 2 2 2

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

128 (1? k )L+ 128 (1? k )L+ 128 (1? k )L; i 6= j 6= k: i jk j ik k ij 3 3 3 The eigenfunctions with bound state energy eigenvalues 1 1 Ep= 2 (2p+ 2+ k+ k+ k )? 8 in these coordinate systems are: polar2 2 2 1 2 3 2

= (sin )? where p= m+ n, (here=( where1

1

(

k2;k1 ) (') (2n+k1+k2;k3 ) ( n m

)2 3

Y3

`=1

s`k`+ 2 ))( 2

(k;k ) n 2 1 is q 1 Y

given by (20)) and elliptical+ s e+ s e j?e j? j?2 1 2 2 1 2 3

s

!

j=1

3

k+1+ k+1+ k+1+ X 2=0 m?e m?e m?e j6 m ( m? j ) and q= p. Here we have made use of the identity s+ s+ s=` (u`? j ): j?e j?e j?e m ( j? em ) The separable eigenfunctions already given are eigenfunctions of the symmetry operators L and e L+ e L+ e L: S= (2n+ k+ k+ 1); E=?2 k (e+ e )+ k (e+ e )+ k (e+ e )+ e k k+ e k k+ e k k]? q 3 (e+ e+ e )? 4e e (k+ 1) X 1? 2 m?e m q q X 1 X 1? 4e e (k+ 1): 4e e (k+ 1) m?e m?e m m A second multiseparable potential on the spher

e is 3 2 (k? ) (k? ) 5 s+ q1 4q V=?q:+q s+s 4 s+s s+s+s s+s?s1 2 3= 2 1 2 2 2 3 1 2 3 2=1 3=1 12 3 12 2 13 1 23 1 2 2 1 2 3 2 1 3 3 2 1 3 1 2 1 2 3 2 1 3 1 2 3 2 3 1=1 1 2 1 3=1 3 1 3 2=1 2 3 2 1 1 4 2 2 1 4 2 1 2 2 2 1 2 2 2 1 2 2 1 2 1 2 2 1

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

Schrodinger's equation for this potential has the form

#@? s@ )+ (s@? s@ )+ (s@? s@ )+ (s@s@s@s@s@s@s 0 2 31 (k? ) (k? ) 5A s+ q1@? q 4q+q=?2E: s+s 4 s+s s+s+s s+s?s This equation admits solution via separation of variables in two coordinates systems: spherical and elliptical coordinates of modified type1 2 2 2 1 1 3 3 2 1 3 2 2 2 3 3 2 1 1 4 2 2 1 4 2 1 2 2 2 1 2 2 2 1 2 2 1 2 1 2 2 1

"

s0= cos f s+ sin f s;1 1 3

s0= s;2 2

s0=? sin f s+ cos f s3 1

3

where and

(y? e?e ) si= (e? ei )(y? e i ); j )(ei j2 1 2 1 2 3 1 1

i; j; k= 1; 2; 3 i 6= j 6= k;1 e= e+ 4 (E? E?);3 2+ 2

v u (e? e ) u sin f= t (e? e );2

1 e= e+ 4 (E+ E? );1 2+ 2 2+ 2+

1 yj= e+ 1 (E+ E?)+ 1 E E? (Zj+ Z ); j= 1; 2: 4 4 j Indeed if we use the variables U= Z Z; U= (Z+(?)(Z1)+? ); U= (Z+( )(Z1)+ )??? where E= E;?= E?; E? q^ then, putting k= 2(E? i )+ 1=4 and E= i+ E and multiplying the?? 1, we see that the resulting equation has Schrodinger equation by (Z Z ) the form@@@@@@ H= (U@U? U@U )+ (U@U? U@U )+ (U@U? U@U ) )+2 3 1 2 2 1 1 2 2 2 2 1+ 2+ 2++++ 3 1 2 1 1 2 2 2 1 1 3 3 2 1 3 2 2 2 3

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

(?k ) (?k ) (?k )^+ 2E= 0 U+ U+ U which is essentially the same form as for the rst potential. The bound state quantisation condition has the form1 4 2 1 2 1 1 4 2 2 2 2 1 4 2 3 2 3

"

#

2 3 s 1 1 E= 2 42q+ 2+ k+ k+ 1+ 2(E? i )5? 8: 42 1 2

4 Features of superintegrability (n=2)Based on the examples of the last two sections, we can point out some basic features of superintegrability in two dimensions. 1. The potential V permits separability of the Hamilton-Jacobi equation H= E and the Schrodinger equation H= E in at least two coordinate systems, characterized by symmetry conditions L=; L= in the rst case and L=; L= in the second. 2. One can obtain alternate spectral resolutions f j g; f k g for the multiply-degenerate eigenspaces of H,1 1 2 2 1 1 2 2 (1) (2)

L j= L k= j; k: These alternate resolutions resolve the degeneracy problem. 3. The interbasis expansions X ajk j k=1 (1) (1) 1 (1) 2 (2) (2) 2 (2) (2) (1)

j

yield important special function identities. In many cases, these become expansions of one set of multivariable orthogonal polynomials in terms of another set. 4. The operators H; L; L generate a quadratic algebra. Indeed, with R= L; L] we have that R is a polynomial of order 3 in H; L; L, whereas L; R] and L; R] are polynomials of order 2 in H; L; L . A corresponding statement is true for algebra generated by the symmetries H; L; L under the Poisson bracket. (Note: This is a remarkable1 2 1 2 2 1 2 1 2 1 2 1 2

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

property, and is fal

se for general symmetries. Consider Euclidean 2space with Hamiltonian H= px+ py . The algebra of all symmetries of H is generated by px; py; M= xpy? ypx. Let2 2

L= M+ px py; L= p x: Then we have R= fL; L g= 4Mpxpy and1 2 2 2 1 2

analytic at this point. Thus it has no power series expansion about the origin.) 5. The quadratic algebra structure can be used to compute the interbase expansion coe cients.

R= F (L; L; L )= 16L L (L? L )? 16L 2 (L? L ) 23: Since H; L; L are functionally independent R must be a function of these symmetries. However, although F is de ned and bounded at the point (L; L; L )= (0; 0; 0), it is not a polynomial, and not even2 0 1 2 1 2 0 2 2 0 2 1 2 2 0 1 2

3

5 Examples in higher dimensionsAn extreme case, superintegrability in n dimensions, occurs for the Schrodinger equation ( n+ Vn(q))=?M (M+ G? 1) (21) where n 1 X ( i? )( i? )? 1 ( n? )( n? )+; (22) Vn=? 4 qi 4 q i1 2=1 2 3 2+1 1 2+1

G= Pn j

+1=1

= 1 (1? G)? 1? (n? 3)(n+ 1); 4 4 j, and n is the Laplace-Beltrami operator on the n-sphere. Here2

"

2 0

3 2

#

q+q+2 0 2 1

+ qn= 1:2

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

One can transfer this Schrodinger equation with a scalar potential Vn to one with vector potential n through the use of a multiplier transformation . Setting (x)= (x) (x) for a nonzero scalar function we nd

() (provided and Here?1

H

n+ n) n+ Vn (x))2 1 4 1

(

=?M (M+ G? 1)=?M (M+ G? 1);

(23)

= x 1=?=n

xnn=?= (1? x) n+1=?=;2 1 4 2 1 4

=

n X j=1

j?

1+ ( n+ 1? G)x]@: j xj 2 2n X i=1

H=

n X

in the coordinates

i;j=1

(xi ij? xi xj )@xixj+

( i? Gxi )@xi

q= 1?2 0

n X i=1

xi= 1? x

q= x q= x ... qn= xn:2 1 2 2 1 2 2

In the paper 17] and the book 18] all separable coordinates for the equation are constructed, where n is the Laplace-Beltrami operator on n= n . It is shown that all separable coordinates are orthogonal and that for S each separable coordinate system the corresponding separated solutions are characterized as simultaneous eigenfunctions of a set of n second order commuting symmetry operators for n. Moreover, the equation ( n+Vn)= where the scalar potential takes the form

Vn=

n X i=1

i qi+ q;0 2 2 0

0

;;:::;1

n

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

is separable in all the coordinate systems in which the Laplace-Beltrami eigenvalue equation is separable. These results can easily be extended to solutions of ( n+ n)= (24) through the mappingsn+ n Sij

===

( n+ Vn )? 0 Sij?:1

1

(25)

Indeed all separable solutions map to R-separable solutions of (24), 21]. Finally, since H= n+ n maps polynomials of maximum order mk in xk to polynomials of the same type, it follows that a basis of separated solutions can be expressed as polynomials in the xi. The second-order symmetry operators for this operator can be chosen to be self-adjoint, so the basis of simultaneous eigenfunctions can be chosen to be orthogonal with respect to the inner product (; )<1 2 1

;

2

>= Z

=1

Z

Z

Z1

xi>0;x<11

1

(x) (x)? (x) d!2 2

(26)

d!= x 1?::: xnn? (1? x) n+1? dx::: dxn:~ Thus every separable coordinate system for the Laplace-Beltrami eigenvalue equation on the n-sphere yields an orthogonal basis of polynomial solutions of equation (24), hence an orthogonal basis for all n-variable polynomials with inner product (26). For details about the bases that can occur and the interbasis expansion coe cients, see 30, 31]. Among the special functions that arise are polyspherical harmonics, products of Jacobi polynomials, Heun polynomials, Lame' polynomials, ellipsoidal polynomials, and Lauricella polynomials. We will look at one more example, the generalized isotropic oscillator in Eucidean 3-space. This is the Schrodinger equation ! 1@+@+@ H=? 2@x@y@z+ V (x; y; z)= E; (27)1 1 1 2 2 2 2 2 2

xi>0;x<1

2

d!;~

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