Jun 4, 1997 - W. Miller, Jr. ..... George for stimulating this interaction and wish them the best on their 60th ... 1] E.G.Kalnins, W.Miller Jr. and G.S.Pogosyan.
Superintegrability on two dimensional complex Euclidean space E. G. Kalnins Department of Mathematics and Statistics, University of Waikato, Hamilton, New Zealand. W. Miller, Jr. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, U.S.A. and G. S. Pogosyan Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region 141980, Russia June 4, 1997
Abstract
In this work we examine the basis functions for those quantum mechanical systems on two dimensional complex Euclidean space, with non zero potential, that admit separation in at least two coordinate systems. We present all of these cases from a uni ed point of view. In particular, all of the polynomial special functions that arise via variable separation have their essential features expressed in terms of their zeros.
1
1 Introduction In a previous work [1] we have examined the basis functions for those classical and quantum mechanical systems in real two dimensional Euclidean space which admit a solution via separation of variables in more than one coordinate system. (We recall that the notion of a superintegrable system relates to a potential for which the solution via separation of variables is possible in more than one coordinate system, [2, 3, 4]. The interest in these systems results from the fact that they have extra integrals of the motion that can be used to break the degeneracy of the energy eigenstates.) The methodical search for such potentials was initiated by Smorodinsky, Winternitz et. al. in two and three dimensions, [5, 6, 7]. (For some related work, see [8, 9, 10, 11].) In this paper we look at the same problem for the case of complex Euclidean space. In so doing we show that there are a number of additional potentials possible other than those of the real case alone. These additional potentials are complexi cations of potentials for the Klein Gordan equation on real at space with inde nite metric. Our basic problem is to solve Schrodinger's equation
H = ? 21 + V (x; y) = ? 21 (@x + @y ) + V (x; y) = E (1) via an ansatz of the form = (u ) (u ) in some suitable coordinate system x = f (u ; u ); y = g(u ; u ) . We are interested in potentials V (x; y) for which this is possible for two or more distinct coordinate systems. For the case of real Euclidean space there are exactly four such potentials and a detailed analysis of their nite solutions or bound states has been given previously, e.g., [1, 8, 10]. We summarize the results of previous work that we need in order to give a readable presentation. In addition to the four potentials possible for real Euclidean space we nd in the complex case three more potentials. For one of the potentials there is the possibility of nite polynomial solutions. We give the details of the eigenfunction solutions and the structure of the quadratic algebra associated with each system. The quadratic algebra yields essential information for the expansion of the basis functions of one separable system in terms of the basis functions for a second system. There are seven possible potentials. The four potentials already treated in the real Euclidean case are. 2
1
1
2
1
1
2
2
2
2
2
k ? k ? (2) V (x; y) = 21 ! (x + y ) + x + y ; k ? V (x; y) = 12 ! (4x + y ) + y ; (3) kp? kp? V (x; y) = p ? + p +p x +y x + y [ x + y + x] x + y [ x + y ? x] (4) qp qp x +y +x x +y ?x V (x; y) = p ? + B p +B p : (5) x +y x +y x +y These potentials are thoroughly treated in our previous article. We now turn our attention to the new potentials that occur in the complex case. The possible orthogonal separable coordinate systems in complex Euclidean space, together with their de ning symmetry operators for the zero potential case [13], are listed in Table 1 and the remaining potentials are listed in Table 2. Here, M = y@x ? x@y ; P = @x ; P = @y : 2
1
2
2
2
4
2
2
2
1 4
2 2
2
1 4
2 1
2
2
1 4
2 2
2
2
3
1 4
2 1
2
2
2
2
2
2
2
2
1
2
1
3
1 4
2 2
2
2
2
2
2
2
2
2
2
2
2
Table 1.
Separable Coordinate Systems on Two-Dimensional Complex Euclidean Space Coordinate System
Integrals of Motion
Coordinates
I. Polar r > 0; 2 [0; ]
I=M
x = r cos y = r sin
II. Cartesian, x; y 2 R
I=P
2
x; y
2 1
III. Light Cone x; y 2 R
I = (P + iP )
IV. Elliptic > 0; 2 [0; 2]
I =M ?a P , a 6= 0
V. Parabolic ; > 0 VI. Hyperbolic
1
2
2
2
x^ = x ? iy y^ = x + iy
2
x = a cosh cos y = a sinh sin
2 2
I = fM; P g
x = ( ? ) y = 1 2
2
2
2
I = M + (P + iP ) x = u22 u2vu2v22 v22 y = i u ?uvuv v 2
1
2
2
(
+
+
2
(
+
) )
2
VII. Semi-Hyperbolic
I = fM; P + iP g +(P ? iP ) 1
1
4
2
2
2
x = ? (w ? z) + (w + z) iy = ? (w ? z) ? (w + z) 1 4 1 4
2
2
1 2 1 2
Table 2.
Complex Superintegrable Systems
Potential V (~x)
Coordinate System
V = B (x ? iy)
Cartesian Light Cone Semi-Hyperbolic
5
2
px +iy
V =
2
V =
1 2
6
7
Parabolic Semi-Hyperbolic Light Cone
h (x2 +y2 ) x iy
( +
)4
+
(x+iy )2
+ (x + y ) 2
2
i
Polar Hyperbolic
2 Potential V5 The Schrodinger equation for potential V admits separable solutions in three distinct coordinate systems. In cartesian coordinates this equation has the form (6) ? 21 (@x + @y ) + 21 B (x ? iy) = E : It admits separable solutions = (x)'(y) which satisfy 5
2
2
@y '(y) ? iyB'(y) = (2E ? )'(y):
@x (x) + Bx (x) = (2E + ) (x); 2
2
The solutions obtained in this way are products of Airy functions [14], typically (x) = Ai (?B ) = x ? 2EB+ ; '(y) = Ai (iB ) = y ? 2EB? : 1 3
1 3
5
The operator K , whose eigenvalue is the separation constant , is de ned by K = (?@x + @y ) + B (x + iy) : In light cone coordinates Schrodinger's equation has the form ? 4@x@y + B y^ = 2E (7) which admits a solution of the form = h(^x)k(^y) where @xh(^x) = h(^x); ?4@y k(^y) + (B y^ ? 2E )k(^y) = 0: These equations have the solution = exp(x^ + 2E y^ ? 8B y^ ): The solutions are clearly eigenfunctions of the operator N = @x + i@y = 2@x: Finally, if we write the Schrodinger equation in semi-hyperbolic coordinates it becomes (@w ? @z ) + B (w ? z ) = 2E (w ? z) : (8) The solutions of this equation can be obtained via the separation of variables ansatz = W (w)Z (z). The separation equations are @x X + Bx X = (2Ex + )X; where X = W; Z when x = w; z, respectively. The solutions of these equations are typically p 1 E 4 X (x) = D 2(?B ) (x ? B ) ; p where = (? ?B )(? ? ( EB ) ) ? , and D (z) is a parabolic cylinder function [14]. The operator, whose eigenvalue is the separation parameter, is L = z ?1 w (z@w ? w@z ) + Bzw: The operators K; L; N and H which de ne these bases satisfy the following quadratic algebra relations [15, 1] : [N; K ] = ? 1 B; [N; L] = ?N ? B; [K; L] = 1 N + fN; K g + 1 BN; (9) 2 2 4 4H + N + 4fN ; K g ? 4BL ? 8K ? BN = 0: Here, [A; B ] = AB ? BA, fA; B g = AB + BA. There are no solutions which are the analogue of bound states in this case. 2
^
2
^
^
^
2
^
2
2
2
2
2
2
1 2
1 2
2
2
2
2
2
4
3
2
2
6
3 Potential V6 For potential V the Schrodinger equation admits separable solutions in three coordinate systems. In parabolic coordinates the Schrodinger equation has the form 6
p
(@ + @ ) ? 2( ? i) = 2E ( + ) : 2
2
2
The separation equations are
p
(10)
2
p
@ X ? 2X = (2E + )X; @ Y + 2iY = (2E ? )Y; 2
2
2
2
where = X ( )Y (). Solutions to these equations are typically 1 1 i 4 4 p p ) ; Y () = D (8E ) ( + ) ; X ( ) = D (8E ) ( + 2E 2E where = p1 (? + 4 E ) ? 12 ; = p1 ( ? 4 E ) ? 12 ; 8E 8E and D (x) is a parabolic cylinder function [14]. The operator whose eigenvalue is is p 1 : i L = + ( @ ? @ ) + +2i 2
2
2
2
2
2
2
2
In light cone coordinates the Schrodinger equation has the form (11) ? 4@x@y + p y^ = 2E ; which admits a solution = p(^x)q(^y) where @xp(^x) = p(^x); ?4@y q(^y) + ( p y^ ? 2E )q(^y) = 0: These equations have the solution q = exp(x^ + 2E y^ ? 2 y^): The solutions are clearly eigenfunctions of the operator N = @x + i@y = 2@x. ^
^
^
^
^
7
For semi-hyperbolic coordinates the Schrodinger equation has the form
p
(@w ? @z ) ? i 2 = 2E (w ? z): 2
2
(12)
The separation equations are @w R(w)+(?2Ew +? pi )R(w) = 0; @z S (z)+(?2Ez ++ pi )S (z) = 0; 2 2 where = R(w)S (z). These equations have Airy function solutions ? i p + i p 1 1 Ai (2E ) 3 w ? 2E ; Ai (2E ) 3 w ? 2E : These solutions are eigenfunctions of the operator 1 (z@ ? w@ ? pi (w + z)): M = w? z z w 2 The basis elements L; M; T and H satisfy the quadratic algebra relations [15, 1] [L; N ] = H +N ; [M; N ] = ? 21 H; [M; L] = 41 N +fN; M g+ 81 HN; (13) HL + 2HM + 18 N H + fN ; M g + 81 N ? 41 = 0: 2
2
2
2
2
2
3
2
2
2
4
2
4 Potential V7
The Schrodinger equation for potential V admits separable solutions in two coordinate systems. In polar coordinates the equation has the form (@r + 1r @r + r1 @ ? r e? i ? r e? i ? r ) = ?2E : (14) The separation equation has the form 7
2
2
2
4
2
2
2 2
2
(@ ? e? i ? e? i ? )() = 0; 2
4
2
8
which has solutions of the form
() = e z2 z? 12 ? 4 ?pLpp 2 ? (? z); where z = e? i , = ?[2p + 1 + ] and Lqp(x) is a Laguerre polynomial with p an integer [14]. The corresponding separation equation in r is (@r + 1r @r + r ? r + 2E )R(r) = 0; which has solutions
2
2 +
1
2
2
2
2 2
2
R(r) = e? 21 r2 r 12 4 pLm2 4 p( r ) for which E = (2(m + p + 1) + ). The operator which corresponds to the eigenvalue is M = @ ? e? i ? e? i : For hyperbolic coordinates the Schrodinger equation has the form [ 1 (v@v (v@v ) ? u@u(u@u)) + ? u + v + (u + v )] = 2E : u ?v uv uv (15) The separation equations are [t@t (t@t ) ? t + t ? t + 2Et + ]T (t) = 0; where = U (u)V (v) and T = U; V when t = u; v, respectively. The operator which characterises this separation is L = u ?1 v (v u@u(u@u)?u v@v (v@v ))+ ( u 1v + u1 + v1 )? ( u1 + v1 )? u v : To obtain polynomial eigenfunctions we can look for solutions of the form [16] = exp [ 1 + 1 ] ? 1 (u + v ) ni ( 1 ? i)( 1 ? i )(uv) n 2 ; 2u v 2 u v where the zeros satisfy X i 2 ? ( + 2n + 2)i + + i = 0; i = 1; ; n: 2 j 6 i i ? j +
1+
+
+
2
2
2
4
2
2
2
2
2
2
2
4
2 4
2
2
2
2
4
2
2
2
2
2
2
2
2
2
2
2
4
2
2
=1
2
2
2
9
4
2
2
1+ +
2
2 2
=
4
2
2
2
The energy eigenvalues and the separation constant are n X E = (2(n + 1) + 2 ); = ?[2n + 1 + 2 ] ? 2 + 4 1 : The relations of the corresponding quadratic algebra are 2
=1
[R; L] = ?8L ? 24M ? 16fL; M g ? 8 H ? 32 ;
(16)
[R; M ] = 16M ? 8M ? 8L + 8 H + 8fL; M g;
(17)
2
2
2
176 L + 80 M R = 16M + 16 ( L M + ML + LML ) ? (18) 3 3 3 ?16 H + 16 (HL + HM ) ? 16fL; M g + (64 ? 256 3 )M 128 H + 64 + 16 ; L + ? 256 3 3 3 where R = [L; M ]. 2
2
2
2
2
2
2
2
5 Acknowledgement Two of the authors of this paper, Kalnins and Miller, met at the CRM in 1973 as a result of invitations from Patera and Winternitz, became involved in the work on variable separation being done at that time, e.g. [12], and have continued collaborating until this day. We are grateful to Pavel and George for stimulating this interaction and wish them the best on their 60th birthdays.
References [1] E.G.Kalnins, W.Miller Jr. and G.S.Pogosyan. Superintegrability and associated polynomial solutions. Euclidean space and the sphere in two dimensions; J.Math.Phys. 37, 6439 (1996). [2] N.W.Evans. Superintegrability in Classical Mechanics; Phys.Rev. A 41 (1990) 5666; Group Theory of the Smorodinsky-Winternitz System; J.Math.Phys. 32, 3369 (1991). 10
[3] S.Wojciechovski. Superintegrability of the Calogero-Moser System. Phys. Lett. A 95, 279 (1983). [4] N.W.Evans. Super-Integrability of the Winternitz System; Phys.Lett. A 147, 483 (1990). [5] J.Fris, V.Mandrosov, Ya.A.Smorodinsky, M.Uhlir and P.Winternitz. On Higher Symmetries in Quantum Mechanics; Phys.Lett. 16, 354 (1965). [6] J.Fris, Ya.A.Smorodinskii, M.Uhlir and P.Winternitz. Symmetry Groups in Classical and Quantum Mechanics; Sov.J.Nucl.Phys. 4, 444 (1967). [7] C.P.Boyer, E.G.Kalnins and P.Winternitz. Completely integrable relativistic Hamiltonian systems and separation of variables in Hermitian hyperbolic spaces J.Math.Phys. 24, 2022 (1983). [8] L.P.Eisenhart. Enumeration of Potentials for Which One-Particle Schrodinger Equations Are Separable; Phys.Rev. 74, 87 (1948). [9] C.Grosche, G.S.Pogosyan, A.N.Sissakian. Path Integral Approach to Superintegrable Potentials. The Two-Dimensional Hyperboloid. Phys. Part. Nucl. 27, 244 (1996). [10] C.Grosche, G.S.Pogosyan, A.N.Sissakian. Path Integral Discussion for Smorodinsky - Winternitz Potentials: I. Two - and Three Dimensional Euclidean Space. Fortschritte der Physik, 43, 453 (1995). [11] P. Letourneau and L. Vinet. Superintegrable Systems: Polynomial Algebras and Quasi-Exactly Solvable Hamiltonians. Ann. Phys., 243, 144168 (1995). [12] J.Patera and P.Winternitz. A New Basis for the Representation of the Rotation Group. Lame and Heun Polynomials; J.Math.Phys. 14, 1130 (1973). [13] W.Miller, Jr. Symmetry and Separation of Variables. Addison-Wesley Publishing Company, Providence, Rhode Island, 1977. [14] A.Erdelyi, W.Magnus, F.Oberhettinger and F.G.Tricomi (eds) Higher Transcendental Functions, Vol. I,II McGraw Hill, New York, 1953. 11
[15] Ya.A.Granovsky, A.S.Zhedanov and I.M.Lutzenko. Quadratic Algebra as a `Hidden' Symmetry of the Hartmann Potential; J.Phys. A 24, 3887 (1991). [16] E.G.Kalnins and W.Miller. Separable coordinates, integrability and the Niven equations. J. Phys. A: Math. Gen. 25, 5663 (1992).
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