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Superintegrability in Three-Dimensional Euclidean Space E View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Research Commons@Waikato JOURNAL OF MATHEMATICAL PHYSICS VOLUME 40, NUMBER 2 FEBRUARY 1999 Superintegrability in three-dimensional Euclidean space E. G. Kalnins and G. C. Williams Department of Mathematics and Statistics, University of Waikato, Hamilton, New Zealand W. Miller, Jr. School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 G. S. Pogosyan Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region 141980, Russia ~Received 19 May 1998; accepted for publication 1 September 1998! Potentials for which the corresponding Schro¨dinger equation is maximally super- integrable in three-dimensional Euclidean space are studied. The quadratic algebra which is associated with each of these potentials is constructed and the bound state wave functions are computed in the separable coordinates. © 1999 American Institute of Physics. @S0022-2488~99!02602-X# I. INTRODUCTION The present paper continues our study of the systems with hidden symmetry or so-called superintegrable systems in spaces with constant curvature. The best known systems of this kind in three-dimensional Euclidean space are the harmonic oscillator and Kepler–Coulomb problems, which have many special properties distinct from other spherically symmetric potentials. These include the phenomena of separation of variables for the Hamilton–Jacobi and Schro¨dinger equations in more than one orthogonal coordinate system and the existence of integrals of motion in addition to the total angular momentum L2. In particular for 2 1 the isotropic oscillator there is the Demkov tensor Dik5pipk1v xixk , and, in the case of the Kepler–Coulomb problem, the Pauli–Runge–Lenz vector A51/2(@L3p#2@p3L#)2r/uru. Both these systems possess five functionally independent integrals of motion.2,3 The first systematic search for all potentials for which the Schro¨dinger equation admits separation of variables in two or more coordinate systems was begun by Smorodinsky and Winternitz with co-workers in Refs. 4–6 and continued by Evans in Refs. 3 and 7. They found all such systems in two- and three- dimensional flat space and introduced the notion of superintegrability. In general, a physical system in N dimensions is called minimally superintegrable if it has 2N22 integrals of motion, and maximally superintegrable if it has 2N21 integral of motions. There are five known maxi- mally ~and some minimally! superintegrable potentials listed in Refs. 3, 8, and 10 and investigated from different points of view in the last decade.8–13 Note also that superintegrable potentials in spaces of constant curvature were introduced in Refs. 14–16. In previous articles17–19 we have looked at potentials in two-dimensional Euclidean space and the two-dimensional sphere and hyperboloid, for which the Schro¨dinger equation is maximally superintegrable. In this article we extend this study to the case of three-dimensional Euclidean space. As previously seen in the case of two dimensions, some of these potentials ~see Table I! admit bound state or finite solutions and it is these to which we draw attention in this article. The basic equation that we investigate is of course the Schro¨dinger equation (\5m51) 1 1 ]2 ]2 ]2 C52 DC1V~x,y,z!C52 1 1 C1V~x,y,z!C5EC. ~1! H 2 2 S]x2 ]y2 ]z2D The idea is to find solutions of this equation via a separation of variables ansatz 0022-2488/99/40(2)/708/18/$15.00708 © 1999 American Institute of Physics Downloaded 23 Oct 2008 to 130.217.76.77. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp J. Math. Phys., Vol. 40, No. 2, February 1999 Kalnins et al. 709 TABLE I. The three-dimensional maximally superintegrable potentials. Potential V(x,y,z) Separating coordinates Cartesian 2 2 1 2 1 2 1 v k12 4 k22 4 k32 4 Spherical V 5 ~x21y21z2!1 1 1 1 2 S x2 y2 z2 D Cylindrical polar Cylindrical elliptic Sphero-conical Oblate spheroidal Prolate spheroidal Ellipsoidal 2 1 2 1 Cartesian v2 1 k 2 k 2 2 2 2 1 4 2 4 Cylindrical polar V25 ~x 1y 14z !1 2 1 2 2 2 S x y D Cylindrical parabolic Cylindrical elliptic Parabolic 1 1 Spheroidal-conical 1 k22 k22 a 1 4 2 4 Spherical V352 1 2 1 2 Ax21y 21z2 2 S x y D Parabolic Prolate spheroidal II 3 C5 c j~u j! j)51 for some suitable orthogonal coordinates u j ~see Table II!. In Secs. II–IV we consider three maximally superintegrable potentials ~see Table I! and use the Niven-type ~or Bethe20! ansatz for constructing the solution of the Schro¨dinger equation in coordinates such as spheroidal, sphero-conical, and ellipsoidal ~see Table II!. In addition we discuss the extension to the quadratic algebras that were in evidence in the case of two dimensions and see what their implications may be. Section V is devoted to the calculation of interbasis expansion coefficients for the V3 potential between spherical and parabolic bases. II. GENERALIZED ISOTROPIC OSCILLATOR The first potential ~see Table I! on our list of three is 2 2 1 2 1 2 1 v 1 ~k12 4 ! ~k22 4 ! ~k32 4 ! V ~x,y,z!5 ~x21y 21z2!1 1 1 , ~2! 1 2 2 F x2 y 2 z2 G 1 1 where the constant ki> 2. For ki5 2 we have the ordinary isotropic oscillator potential. The corresponding Schro¨dinger equation admits solutions via a separation of variables in eight coor- dinate systems: Cartesian, spherical, sphero-conical, cylindrical polar, cylindrical elliptic, prolate and oblate spheroidal, and ellipsoidal. We summarize the bound state solutions in each case. Before considering various coordinate systems we note that a basis for the symmetries of Schro¨dinger’s equation with the potential ~2! consists of the six operators: Downloaded 23 Oct 2008 to 130.217.76.77. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp 710 J. Math. Phys., Vol. 40, No. 2, February 1999 Kalnins et al. TABLE II. Systems of coordinate in three-dimensional Euclidean space. Coordinate system Coordinates I. Cartesian x,y,z x,y,zPR II. Cylindrical polar x5r cos w, y5r sin w, z r.0, wP@0,2p) III. Cylindrical elliptic ~m12e1!~m22e1! ~m12e2!~m22e2! x25 , y25 , z zPR, e1,m1,e2,m2 ~e22e1! ~e12e2! IV. Cylindrical parabolic 1 2 2 x,y5jh, z5 2(j 2h ) j,xPR, h>0 V. Spherical x5r cos u cos w, y5r sin u sin w, z5r cos u r.0, uP@0,p#, wP@0,2p) ~u12e2!~u22e2! VI. Prolate spheroidal x25 cos2 w, ~e 2e ! e1,u1,e2,u2 , wP@0,2p) 1 2 ~u12e2!~u22e2! y25 sin2 w, ~e12e2! ~u12e1!~u22e1! z25 ~e22e1! ~u12e1!~u22e1! VII. Oblate spheroidal x25 cos2 w, ~e 2e ! e1,u1,e2,u2 , wP@0,2p) 2 1 ~u12e1!~u22e1! y25 sin2 w, ~e22e1! ~u12e2!~u22e2! z25 ~e12e2! ~r12e1!~r22e1! VIII. Sphero-conical x25r2 , ~e 2e !~e 2e ! r>0, e1,r1,e2,r2,e3 1 2 1 3 ~r12e2!~r22e2! y25r2 , ~e22e1!~e22e3! ~r12e3!~r22e3! z25r2 ~e32e2!~e32e1! IX. Parabolic 1 2 2 x5jh cos w, y5jh sin w, z5 2(j 2h ) j,h>0, wP@0,2p) ~u12a1!~u22a1!~u32a1! X. Ellipsoidal x25 , ~a 2a !~a 2a ! a1,u1,a2,u2,a3,u3 3 1 2 1 ~u12a2!~u22a2!~u32a2! y25 , ~a12a2!~a32a2! ~u12a3!~u22a3!~u32a3! z25 ~a12a3!~a22a3! XI. Paraboloidal 0,h ,a ,h 1 2 2 ~h12a3!~h22a3!~h32a3! 25 ,a3,h3 x , ~a32a2! ~h12a2!~h22a2!~h32a2! y25 , ~a22a3! 2 1 z 5 2(h11h21h32a22a3) Downloaded 23 Oct 2008 to 130.217.76.77. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp J. Math. Phys., Vol. 40, No. 2, February 1999 Kalnins et al. 711 2 1 ki 2 4 M i52Dii2 2 , 2 5M11M21M3 , ~3! xi H 2 2 1 xj 1 xi 1 J 5L2 2 k22 2 k22 2 , i,j51,2,3, ~4! ij ij S i 4Dx2 S j 4Dx2 2 i j where L 5x ] 2x ] , D 52]2 1v2x2 is a diagonal components of the Demkov tensor1 and ij i xj j xi ii xi i we have the notation x15x, x25y, x35z. The commutators of the operators ~3! and ~4! can be closed to form a quadratic algebra as follows: @M i , M j#50, @M i , J jk#50, @M i , Jij#5Qij5Q@ij# , @Jij ,Jik#5R@ijk#5R, where Qij is totally antisymmetric and the totally antisymmetric quantity R@ijk# is denoted by R. Further commutators are calculated to be @M i , Q jk#50, @M i , Qij#54$Mi ,Mj%116Jij , @Mi ,R#54$Mk ,Jij%24$Mj ,Jik%, 2 2 @Jij ,Qij#54$Mi ,Jij%24$Mj ,Jij%28~kj21!Mi18~ki 21!Mj , @Jij ,Qik#54$Mi ,Jjk%24$Mj ,Jik%, 2 2 @Jij ,R#54$Jij ,Jjk%24$Jij ,Jik%28~ki 21!Jjk18~kj 21!Jik , where $A,B%5AB1BA. The expression for the commutators of the Q and R are @Qij ,Qik#54$Mi ,Qjk%, @Qij ,R#524$Jij ,Qik%24$Jij ,Qjk%. All the commutators of the operators M i , Jmn , Qpq , and R can be expressed in terms of quadratic symmetric products of themselves. The algebra, therefore, is closed quadratically. There are rela- tions between the symmetric products of the generators of this algebra. The exhaustive list of these is as follows: 2 8 64 2 2 2 2 Qij53$Jij ,Mi ,Mj%1 3 $Mi ,Mj%116v Jij216~12k j !M i 2 2 128 2 2 2 2 216~12ki !M j 2 3 v Jij264v ~12ki !~12k j !, 8 8 8 $Qij ,Qik%5 3$Jij ,Mi ,Mk%13$Jik ,Mi ,Mj%2 3$Jjk ,Mi ,Mi% 2 2 2 2 2 132v ~12ki !$Jij ,Jik%232~12ki !M jM k264v ~12ki !J jk , 8 8 8 64 64 $Qij ,R%53$Jij ,Jij ,Mk%23$Jij ,Jik ,Mj%2 3$Jij ,Jjk ,Mi%2 3 $Jij ,Mk%2 3 $Jik ,Mj% 64 2 2 2 2 2 3 $Jjk ,Mi%116~12ki !$J jk ,Mj%116~12k j !$Jik ,Mi%264~12ki !~12k j !M k , 2 4 64 64 64 2 2 R 52 3$Jij ,Jik ,Jjk%1 3 $Jij ,Jik%1 3 $Jij ,Jjk%1 3 $Jik ,Jjk%216~12kk !Jij 2 2 2 2 128 2 216~12k j !Jik216~12ki !J jk1 3 ~12kk!Jij 128 2 128 2 2 2 2 1 3 ~12kj !Jik1 3 ~12ki !Jjk164~12ki !~12k j !~12kk !, where $A,B,C%5ABC1CAB1BCA.
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