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New Families of Exactly Solvable Many-Body Models and Superintegrable Systems

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New Families of Exactly Solvable Many-Body Models and Superintegrable Systems New families of exactly solvable many-body models and superintegrable systems Zhe Chen M.Sc. A thesis submitted for the degree of Master of Philosophy at The University of Queensland in 2019 School of Mathematics and Physics ii Abstract In this thesis, we study various extensions of Calogero models and superintegrable systems. We construct a kN-body one-dimensional model which reduces to the familiar Calogero model when k = 1. We present a class of many-body systems that are equivalent to harmonic oscillators. We study interesting extensions of the D-dimensional Coulomb-Kepler system and show that when the extension satisfies certain conditions, then some components of the Laplace-Runge-Lenz vector can be extended to conserved quantity of the new models. By introducing block separation of variables, we construct the Kepler-singular oscillator type models which are a new family of superintegrable systems. We also use separation of variables to obtain the energy spectrum, eigenfunctions and corresponding quadratic algebraic structures. Separation of variables is not only used for solving eigenvalue problems but also provides us with new tools for generalizing superintegrable systems. We generalize the double harmonic-singular oscillators and Kepler-singular oscillator systems. The integrals of new models now can involve angles from block spherical coordinates. A derivation of more general quadratic algebraic structures is presented as well. We also give examples to show they can be solved in terms of so-called X1 Jacobi polynomials. ABSTRACT iii Declaration by author This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, financial support and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my higher degree by research candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis and have sought permission from co-authors for any jointly authored works included in the thesis. iv Publications included in this thesis 1. Chen, Z., Links, J., Marquette, I., & Zhang, Y. Z. (2018). Extended Calogero models: a construction for exactly solvable kN-body systems. Journal of Physics A: Mathematical and Theoretical, 51(45), 455203. Contributor Zhe Chen Jon Links Ian Marquette Yao Zhong Zhang Conception and design 25% 25% 25% 25% Analysis and interpretation 25% 25% 25% 25% Drafting and production 25% 25% 25% 25% 2. Chen, Z. (2018). Mapping quantum many-body system to decoupled harmonic oscillators: General discussions and examples. Physics Letters A, 382(37), 2613-2617. Contributor Zhe Chen Conception and design 100% Analysis and interpretation 100% Drafting and production 100% 3. Chen, Z. (2018). Two Algorithms for Constructing Solvable Quantum N · 2k and N · 3k-Body Models. Few-Body Systems, 59(6), 126. Contributor Zhe Chen Conception and design 100% Analysis and interpretation 100% Drafting and production 100% 4. Chen, Z., Marquette, I., & Zhang, Y. Z. (2019). Extended Laplace-Runge-Lenz vectors, new family of superintegrable systems and quadratic algebras. Annals of Physics, 402, 78-90. Contributor Zhe Chen Ian Marquette Yao Zhong Zhang Conception and design 40% 30% 30% Analysis and interpretation 40% 30% 30% Drafting and production 40% 30% 30% 5. Chen, Z., Marquette, I., & Zhang, Y. Z. (2019). Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras. arXiv preprint arXiv:1905.00194. ABSTRACT v Contributor Zhe Chen Ian Marquette Yao Zhong Zhang Conception and design 40% 30% 30% Analysis and interpretation 40% 30% 30% Drafting and production 40% 30% 30% Publications during candidature Peer-reviewed publications 1. Chen, Z., Links, J., Marquette, I., & Zhang, Y. Z. (2018). Extended Calogero models: a construction for exactly solvable kN-body systems. Journal of Physics A: Mathematical and Theoretical, 51(45), 455203. 2. Chen, Z. (2018). Mapping quantum many-body system to decoupled harmonic oscillators: General discussions and examples. Physics Letters A, 382(37), 2613-2617. 3. Chen, Z. (2018). Two Algorithms for Constructing Solvable Quantum N · 2k and N · 3k-Body Models. Few-Body Systems, 59(6), 126. 4. Chen, Z., Marquette, I., & Zhang, Y. Z. (2019). Extended Laplace-Runge-Lenz vectors, new family of superintegrable systems and quadratic algebras. Annals of Physics, 402, 78-90. Online e-print publications 1. Chen, Z., Marquette, I., & Zhang, Y. Z. (2019). Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras. arXiv:1905.00194. Contributions by others to the thesis My supervisors A/Prof. Yao-Zhong Zhang, A/Prof. Jon Links and Dr. Ian Marquette were responsible for reviewing the thesis. In this capacity, they have provided constructive and timely feedback which have substantially improved the quality of the thesis. Statement of parts of the thesis submitted to qualify for the award of another degree No works submitted towards another degree have been included in this thesis. Research involving human or animal subjects No animal or human subjects were involved in this research. vi Acknowledgments I wish to express my deepest gratitude to my principal supervisor Associate Professor Yao-Zhong Zhang, associate supervisors Associate Professor Jon Links and Dr. Ian Marquette for all their professional guidance, constructive feedbacks and motivations throughout my graduate study at the University of Queensland. They introduced me to the interesting and vibrant fields of (quasi-)exactly solvable models, superintegrable systems, symmetry algebras and their connection to mathematics and physics. They have dedicated their time, knowledge and skills to guide and help me in my study. This thesis would not be possible without their help. I would like to thank the Australian Government and the University of Queensland for the Aus- tralian Government Research Training Program Scholarship. ABSTRACT vii Financial support This research was supported by an Australian Government Research Training Program Scholarship. Keywords Exactly-solvable systems, Superintegrable systems, Symmetery algebras. Australian and New Zealand Standard Research Classifications (ANZSRC) 010501 Algebraic Structures in Mathematical Physics 40% 010503 Mathematical Aspects of Classical Mechanics, Quantum Mechanics and Quantum Information Theory 60% Fields of Research (FoR) Classification FoR code: 0105 Mathematical Physics 100% Contents Abstract............................................ ii Contents viii 1 Introduction 1 1.1 Background....................................... 1 1.2 Aims of research .................................... 4 1.3 Structure of thesis.................................... 4 2 Quantum systems7 2.1 Exactly solvable systems................................ 7 2.1.1 Exact solvability ................................ 7 2.1.2 Separation of variables............................. 8 2.1.3 Calogero systems................................ 13 2.2 Superintegrable systems................................. 18 2.2.1 Integrability and superintegrability....................... 18 2.2.2 Algebraic structures in superintegrable systems................ 18 2.2.3 Qudratic algebras for an extended Kepler-Coulomb problem......... 19 3 Extended Calogero models: a construction for exactly solvable kN-body systems 23 3.1 Introduction....................................... 23 3.2 General discussion and results ............................. 24 3.2.1 A type Calogero model............................. 26 3.2.2 BC type Calogero model ............................ 26 3.2.3 E8 type Calogero model ............................ 27 3.2.4 F4 type Calogero model............................. 27 3.2.5 G2 type Calogero model ............................ 28 3.3 Construction of new models .............................. 28 3.3.1 A type coupling................................. 30 3.3.2 BC type coupling................................ 30 3.3.3 E8 type coupling ................................ 30 3.3.4 F4 type coupling ................................ 31 viii CONTENTS ix 3.3.5 G2 type coupling ................................ 31 3.4 Exactly solvable 3N-body problems .......................... 31 3.5 Conclusion ....................................... 34 4 Equivalence between a class
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