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 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording orotherwise,withoutthepriorpermissionofthepublisher,orasexpresslypermittedbylawor undertermsagreedwiththeappropriaterightsorganization.Multiplecopyingispermittedin accordancewiththetermsoflicencesissuedbytheCopyrightLicensingAgency,theCopyright ClearanceCentreandotherreproductionrightsorganisations. PermissiontomakeuseofIOPPublishingcontentotherthanassetoutabovemaybesought [email protected]. ErnestGKalnins,JonathanMKressandWillardMillerJrhaveassertedtheirrighttobeiden- tifiedastheauthorsofthisworkinaccordancewithsections77and78oftheCopyright,Designs andPatentsAct1988. 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) BritishLibraryCataloguing-in-PublicationData:Acataloguerecordforthisbookisavailable fromtheBritishLibrary. PublishedbyIOPPublishing,whollyownedbyTheInstituteofPhysics,London IOPPublishing,TempleCircus,TempleWay,Bristol,BS16HG,UK USOffice:IOPPublishing,Inc.,190NorthIndependenceMallWest,Suite601,Philadelphia, PA19106,USA 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