title: A recurrence relation approach to higher order quantum superintegrability

authors: Kalnins, Ernie G.

subjects: superintegrability

abstract: We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous proofs of superintegrability and explicit constructions of higher order generators for the symmetry algebra. We apply the method to 5 families of systems, each depending on a parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system, and we give proofs of quantum superintegrability for all rational values of k, new for 4 of these systems. In addition, we show that the explicit information supplied by the special function recurrence relations allows us to prove, for the first time in 4 cases, that the symmetry algebra generated by our lowest order symmetries closes and to determine the associated structure equations of the algebras for each k. We have no proof that our generating symmetries are of lowest possible order, but we have no counterexamples, and we are confident we can can always find any missing generators from our raising and lowering operator recurrences. We also get for free, one variable models of the action of the symmetry algebra in terms of difference operators. We describe how the Stäckel transform acts and show that it preserves the structure equations.

text: A recurrence relation approach to higher order quantum superintegrability We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous proofs of superintegrability and explicit constructions of higher order generators for the symmetry algebra. We apply the method to 5 families of systems, each depending on a parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system, and we give proofs of quantum superintegrability for all rational values of k, new for 4 of these systems. In addition, we show that the explicit information supplied by the special function recurrence relations allows us to prove, for the first time in 4 cases, that the symmetry algebra generated by our lowest order symmetries closes and to determine the associated structure equations of the algebras for each k. We have no proof that our generating symmetries are of lowest possible order, but we have no counterexamples, and we are confident we can can always find any missing generators from our raising and lowering operator recurrences. We also get for free, one variable models of the action of the symmetry algebra in terms of difference operators. We describe how the Stäckel transform acts and show that it preserves the structure equations.

year: 2011

title: Malleable boundaries - Creative modes of practice and inquiry in the Academy

authors: Hill, Rodrigo

abstract: The exhibition ‘Malleable Boundaries’ considers the burgeoning field of creative and non-traditional research outputs across our academic disciplines at the University of Waikato. The show highlights work from practicing artists, emerging and established academics, all testing the malleability of the academy's boundaries and challenging our collective understanding of what research can look like in practice.

Creativity and non-traditional research approaches have a strong footing across disciplines, at times transdisciplinary and interdisciplinary, often transcending traditional boundaries. Discover how creative thinking thrives across our faculties and schools, from the School of Arts and Faculty of Māori and Indigenous Studies, through Education and Design, to the unexpected realms of Law and the Sciences.

Across the exhibition are practicing artists working in photography and paint, weaving, audio/visual and performance. Alongside them are qualitative researchers that have ventured into non-traditional outputs including cartography, documentary-based production, game design, comic book creation and posters. Join us in recognizing the innovative spirit that defines our academic community here at the University of Waikato.

text: Malleable boundaries - Creative modes of practice and inquiry in the Academy The exhibition ‘Malleable Boundaries’ considers the burgeoning field of creative and non-traditional research outputs across our academic disciplines at the University of Waikato. The show highlights work from practicing artists, emerging and established academics, all testing the malleability of the academy's boundaries and challenging our collective understanding of what research can look like in practice.

Creativity and non-traditional research approaches have a strong footing across disciplines, at times transdisciplinary and interdisciplinary, often transcending traditional boundaries. Discover how creative thinking thrives across our faculties and schools, from the School of Arts and Faculty of Māori and Indigenous Studies, through Education and Design, to the unexpected realms of Law and the Sciences.

Across the exhibition are practicing artists working in photography and paint, weaving, audio/visual and performance. Alongside them are qualitative researchers that have ventured into non-traditional outputs including cartography, documentary-based production, game design, comic book creation and posters. Join us in recognizing the innovative spirit that defines our academic community here at the University of Waikato.

year: 2024

title: Waiwharangi

authors: Maxwell, Te Kahautu

abstract: Haka. Lyrics/Musical score composed by Te Kahautu Maxwell. Performed by Ōpōtiki Mai Tawhiti (Kapa Haka group) (4mins)

Waiata: Tira Aroha
Whakaeke: Haere mai rā
Mōteatea: Te Pātere a Te Turuki
Poi: Iarararararī
Waiata ā ringa: Taku Mana Māori Motuhake
Haka: Waiwharangi
Whakawātea: Āio

text: Waiwharangi Haka. Lyrics/Musical score composed by Te Kahautu Maxwell. Performed by Ōpōtiki Mai Tawhiti (Kapa Haka group) (4mins)

Waiata: Tira Aroha
Whakaeke: Haere mai rā
Mōteatea: Te Pātere a Te Turuki
Poi: Iarararararī
Waiata ā ringa: Taku Mana Māori Motuhake
Haka: Waiwharangi
Whakawātea: Āio

year: 2020-02-06