Separation of Variables and Superintegrability: The Symmetry of Solvable Systems

325 Pages·2018·5.736 MB·English

by  Ernest G. Kalnins

Separation 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
Version:20180501
IOPExpandingPhysics
ISSN2053-2563(online)
ISSN2054-7315(print)
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PublishedbyIOPPublishing,whollyownedbyTheInstituteofPhysics,London
IOPPublishing,TempleCircus,TempleWay,Bristol,BS16HG,UK
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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