Table Of ContentSeparation of Variables and
Superintegrability
The symmetry of solvable systems
Separation of Variables and
Superintegrability
The symmetry of solvable systems
Ernest G Kalnins
The University of Waikato, Hamilton, New Zealand
Jonathan M Kress
The University of New South Wales, Sydney, Australia
Willard Miller Jr
University of Minnesota, Minneapolis, USA
IOP Publishing, Bristol, UK
ªIOPPublishingLtd2018
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ISBN 978-0-7503-1314-8(ebook)
ISBN 978-0-7503-1315-5(print)
ISBN 978-0-7503-1316-2(mobi)
DOI 10.1088/978-0-7503-1314-8
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IOPExpandingPhysics
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ISSN2054-7315(print)
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Contents
Preface x
Acknowledgment xiv
Author biographies xv
1 Introduction 1-1
References 1-4
2 Background and definitions 2-1
2.1 Classical mechanics 2-1
2.2 Quantum mechanics 2-5
2.3 Integrability and superintegrability 2-7
2.3.1 Classical integrability and superintegrability 2-8
2.3.2 Extension to quantum systems 2-9
References 2-10
3 Separation of variables 3-1
3.1 Some approaches to separability 3-1
3.1.1 The intuitive concept 3-1
3.1.2 The Fourier approach 3-2
3.1.3 The mathematics of Stäckel form 3-2
3.1.4 The Stäckel procedure for operator equations 3-5
3.2 The Levi-Civita procedure 3-8
3.2.1 Levi-Civita procedure implies Stäckel structure 3-11
3.3 Nonorthogonal separation: examples 3-14
3.4 Intrinsic characterization of separation 3-20
3.4.1 Orthogonal N-tuples and coefficients of rotation 3-21
3.4.2 Intrinsic characterization of Helmholtz separability 3-24
3.4.3 Intrinsic characterization of Laplace separability 3-27
References 3-35
4 Side condition separation 4-1
4.1 A generalization of Stäckel form 4-1
4.2 Generalized Helmholtz Stäckel form 4-4
4.3 Maximal non-regular separation 4-5
4.3.1 Maximal separation implies generalized Stäckel form 4-6
v
SeparationofVariablesandSuperintegrability
4.4 Examples of non-regular separability 4-8
4.4.1 Examples of restricted regular separation 4-8
4.4.2 Non-regular separation in 2D and a ‘no go’ theorem 4-11
4.4.3 Non-regular separation in more than two dimensions 4-12
References 4-17
5 Separation for the real n-sphere 5-1
5.1 Jacobi elliptic coordinates 5-1
5.1.1 Construction of all separable systems 5-4
5.1.2 Separable systems as limits of generic systems 5-7
5.2 Killing vectors and tensors 5-7
5.2.1 Characterization of ellipsoidal coordinates 5-13
5.2.2 Comments and references 5-14
References 5-14
6 Separation for real Euclidean n-space 6-1
6.1 Elliptic coordinates in Euclidean space 6-1
6.2 Parabolic coordinates in Euclidean space 6-3
6.3 Construction of all separable coordinates 6-4
6.4 Comments and references 6-6
References 6-7
7 Separation on the hyperboloid 7-1
7.1 Branching rules for hyperbolic n-space 7-3
7.2 Separation for hyperbolic three-space 7-4
References 7-7
8 Conformally flat spaces 8-1
8.1 Hyperspherical coordinates 8-2
8.2 Separable coordinates: analytic theory 8-4
8.2.1 Construction of separable coordinates 8-8
8.3 Separable coordinates: algebraic theory 8-10
8.3.1 Type I coordinates 8-16
8.3.2 Type II coordinates as limits 8-20
8.3.3 Branching rules for type II coordinates on complex 8-22
nD constant curvature spaces.
8.3.4 All separable systems on 2D complex Euclidean space 8-24
vi
SeparationofVariablesandSuperintegrability
8.3.5 Separable coordinates on the complex two-sphere 8-26
8.3.6 Separable coordinates on the complex three-sphere 8-27
8.3.7 All separable systems on complex 3D Euclidean space 8-33
8.3.8 ‘Real’ cyclides 8-37
8.4 Comments and references 8-41
References 8-43
9 Time-dependent equations 9-1
9.1 Case (i): time as ignorable variable 9-6
9.2 Case (ii): time-dependent Hamiltonians 9-8
9.3 Coordinates on spheres and Euclidean spaces 9-12
9.4 Examples 9-18
References 9-20
10 Generalized Lie symmetries 10-1
References 10-5
11 Differential Stäckel form 11-1
11.0.1 D-Stäckel matrices 11-3
11.0.2 Analysis of the separation equations 11-5
11.1 Separation of Laplace equations 11-9
References 11-10
12 Functional separation 12-1
12.1 A forced wave equation 12-1
12.2 Pseudo-Riemannian spaces 12-6
References 12-10
13 Vector equations 13-1
13.0.1 Spinor form of the Maxwell equations 13-3
13.0.2 Generalized Hertz potentials 13-5
13.0.3 Toward a general theory 13-6
13.1 Dirac-type equations 13-8
13.1.1 Factorisable systems 13-11
13.1.2 Dirac equations as factorisable systems 13-15
vii
SeparationofVariablesandSuperintegrability
13.1.3 A counterexample 13-18
13.1.4 Related work 13-19
References 13-19
14 Links with r-matrix theory 14-1
14.1 Complex constant curvature spaces 14-1
14.2 Generic ellipsoidal coordinates 14-5
14.3 Cyclidic coordinates 14-12
14.3.1 Quantum constant curvature systems 14-17
References 14-21
15 Multiseparability 15-1
15.0.1 Some instructive examples 15-2
15.1 2D superintegrable systems 15-5
15.1.1 Quadratic algebras 15-8
15.1.2 The Stäckel transform 15-12
15.1.3 Laplace equations and conformal superintegrability 15-15
15.1.4 Bôcher contractions 15-17
15.1.5 Composition of Bôcher contractions 15-21
15.1.6 Exact and quasi-exact solvability 15-24
15.1.7 2D Laplace superintegrable systems 15-30
15.1.8 Bôcher contractions 15-31
15.1.9 Stäckel transforms 15-36
15.1.10 Limits of separable coordinate systems 15-41
15.1.11 The separation equations 15-42
15.1.12 Special functions 15-45
15.1.13 More on QES 15-45
15.1.14 Explicitly solvable QES 1D systems 15-46
15.1.15 2D summary 15-49
15.2 Canonical equations 15-49
15.2.1 The wave equation and Gaussian hypergeometric 15-50
functions
15.2.2 Superintegrable canonical equations 15-51
15.3 3D superintegrable systems 15-53
15.3.1 ‘Generic’ Euclidean superintegrable systems 15-56
15.3.2 ‘Generic’ three-sphere superintegrable systems 15-59
viii
SeparationofVariablesandSuperintegrability
15.3.3 3D classical Laplace systems 15-62
15.3.4 3D Bôcher contractions 15-67
15.3.5 Application of the Bôcher contraction 15-72
15.3.6 3D quantum systems 15-74
15.3.7 3D Laplace operator equations 15-76
15.4 Conclusions and extensions 15-77
References 15-78
ix
