I obtained a Ph.D in physics in 2009 from the Université de Montréal. I also obtained a FQRNT fellowship and spend two years in England. More recently, I obtained in 2013 a Discovery early career award from the ARC and in 2018 a Future Fellowship.
My research lie in the field of mathematical physics. I am interested by integrable exactly solvable systems, their related algebraic structures and special functions.
1.Integrable, superintegrable and exactly solvable models, related differential equations and algebraic structures
2.Lie, quadratic and polynomial Lie algebras, realizations, indecomposable representations
3.Casimir invariants, construction and applications, non-semi simple Lie algebras
4.Algebraic Bethe Ansatz, quantum inverse scattering method and phase transitions
5.Painlevé transcendents, exceptional orthogonal polynomials and relation to quantum mechanics
Journal Article: Polynomial algebras from Lie algebra reduction chains g ⊃ g ′
Campoamor-Stursberg, Rutwig, Latini, Danilo, Marquette, Ian and Zhang, Yao-Zhong (2023). Polynomial algebras from Lie algebra reduction chains g ⊃ g ′. Annals of Physics, 459 169496, 1-19. doi: 10.1016/j.aop.2023.169496
Journal Article: Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces
Marquette, Ian, Zhang, Junze and Zhang, Yao-Zhong (2023). Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces. Journal of Physics A: Mathematical and Theoretical, 56 (35) 355201, 355201. doi: 10.1088/1751-8121/ace949
Journal Article: The Lie algebra of the lowest transitively differential group of degree three
Grundland, Alfred Michel Michel and Marquette, Ian (2023). The Lie algebra of the lowest transitively differential group of degree three. Journal of Physics A: Mathematical and Theoretical, 56 (34) 345205, 345205. doi: 10.1088/1751-8121/ace866
Representation theory in exactly solvable systems
(2019–2024) ARC Future Fellowships
New constructions of superintegrable systems and the connection with Painleve transcendents
(2013–2016) ARC Discovery Early Career Researcher Award
ResTeach Funding 2012 0.1 FTE School of Math & Physics
(2012–2013) UQ ResTeach
Quadratic Algebras and Their Casimir Invariant
Doctor Philosophy
Superintegrable systems with Dunkl operators, their symmetry algebras and invariant
Doctor Philosophy
Superintegrabe systems in curved spaces and algebraic approaches
Doctor Philosophy
Construction of Casimir operators of higher rank quadratic algebras
This project will intend to make new discoveries in regard of quadratic algegbras which naturally occur in context of integrable systems and to develop method to develop to obtain their Casimir operators. These results will allow to make further progress in their representation theory and obtain the spectrum of new superintegrable systems.
Indecomposable representations and realizations related to exactly solvable systems
This project will intend to develop differential operator realizations connected to indecomposable representations and apply to the construction of exactly solvable systems.
Marquette, Ian and Winternitz, Pavel (2019). Higher order quantum superintegrability: a new “Painlevé Conjecture”: higher order quantum superintegrability. Integrability, supersymmetry and coherent states: A Volume in Honour of Professor Véronique Hussin. (pp. 103-131) edited by Şengül Kuru, Javier Negro and Luis M. Nieto. Cham, Switzerland: Springer International Publishing. doi: 10.1007/978-3-030-20087-9_4
Polynomial algebras from Lie algebra reduction chains g ⊃ g ′
Campoamor-Stursberg, Rutwig, Latini, Danilo, Marquette, Ian and Zhang, Yao-Zhong (2023). Polynomial algebras from Lie algebra reduction chains g ⊃ g ′. Annals of Physics, 459 169496, 1-19. doi: 10.1016/j.aop.2023.169496
Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces
Marquette, Ian, Zhang, Junze and Zhang, Yao-Zhong (2023). Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces. Journal of Physics A: Mathematical and Theoretical, 56 (35) 355201, 355201. doi: 10.1088/1751-8121/ace949
The Lie algebra of the lowest transitively differential group of degree three
Grundland, Alfred Michel Michel and Marquette, Ian (2023). The Lie algebra of the lowest transitively differential group of degree three. Journal of Physics A: Mathematical and Theoretical, 56 (34) 345205, 345205. doi: 10.1088/1751-8121/ace866
Non-Hermitian superintegrable systems
Correa, Francisco, Inzunza, Luis and Marquette, Ian (2023). Non-Hermitian superintegrable systems. Journal of Physics A: Mathematical and Theoretical, 56 (34) 345207, 345207. doi: 10.1088/1751-8121/ace506
Infinite-dimensional representations of cubic and quintic algebras and special functions
Marquette, Ian, Zhang, Junze and Zhang, Yao-Zhong (2023). Infinite-dimensional representations of cubic and quintic algebras and special functions. European Physical Journal Plus, 138 (6) 537. doi: 10.1140/epjp/s13360-023-04155-2
Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras
Campoamor-Stursberg, Rutwig, Latini, Danilo, Marquette, Ian and Zhang, Yao-Zhong (2023). Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras. Journal of Physics A: Mathematical and Theoretical, 56 (4) 045202, 1-27. doi: 10.1088/1751-8121/acb576
Marquette, I. and Zelaya, K. (2022). On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials. Physica D: Nonlinear Phenomena, 442 133529, 1-10. doi: 10.1016/j.physd.2022.133529
Marquette, I. and Quesne, C. (2022). Algebraic construction of associated functions of nondiagonalizable models with anharmonic oscillator complex interaction. Reports on Mathematical Physics, 90 (3), 285-298. doi: 10.1016/s0034-4877(22)00077-5
Generalized quadratic commutator algebras of PBW-type
Marquette, Ian, Yates, Luke and Jarvis, Peter D. (2022). Generalized quadratic commutator algebras of PBW-type. Journal of Mathematical Physics, 63 (12) 121703, 1-23. doi: 10.1063/5.0096769
Latini, Danilo, Marquette, Ian and Zhang, Yao-Zhong (2022). Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators. Physica D: Nonlinear Phenomena, 440 133464, 1-20. doi: 10.1016/j.physd.2022.133464
Dynamical symmetry algebras of two superintegrable two-dimensional systems
Marquette, Ian and Quesne, Christiane (2022). Dynamical symmetry algebras of two superintegrable two-dimensional systems. Journal of Physics A: Mathematical and Theoretical, 55 (41) 415203. doi: 10.1088/1751-8121/ac9164
Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
Latini, Danilo, Marquette, Ian and Zhang, Yao-Zhong (2022). Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras. Journal of Physics A: Mathematical and Theoretical, 55 (33) 335203, 1-42. doi: 10.1088/1751-8121/ac7ca3
A family of fourth-order superintegrable systems with rational potentials related to Painlevé VI
Marquette, I., Post, S. and Ritter, L. (2022). A family of fourth-order superintegrable systems with rational potentials related to Painlevé VI. Journal of Physics A: Mathematical and Theoretical, 55 (15) 155201. doi: 10.1088/1751-8121/ac550a
Campoamor-Stursberg, Rutwig and Marquette, Ian (2022). Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras. Annals of Physics, 437 168694, 168694. doi: 10.1016/j.aop.2021.168694
Marquette, Ian and Quesne, Christiane (2022). Ladder operators and hidden algebras for shape invariant nonseparable and nondiagonalizable models with quadratic complex interaction. I. two-dimensional model. Symmetry, Integrability and Geometry: Methods and Applications, 18 004. doi: 10.3842/sigma.2022.004
Marquette, Ian and Quesne, Christiane (2022). Ladder operators and hidden algebras for shape invariant nonseparable and nondiagonalizable models with quadratic complex interaction. II. three-dimensional model. Symmetry, Integrability and Geometry: Methods and Applications, 18 005, 1-24. doi: 10.3842/sigma.2022.005
Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials
Hussin, Veronique, Marquette, Ian and Zelaya, Kevin (2022). Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials. Journal of Physics A: Mathematical and Theoretical, 55 (4) 045205. doi: 10.1088/1751-8121/ac43cc
Racah algebra R(n) from coalgebraic structures and chains of R(3) substructures
Latini, Danilo, Marquette, Ian and Zhang, Yao-Zhong (2021). Racah algebra R(n) from coalgebraic structures and chains of R(3) substructures. Journal of Physics A: Mathematical and Theoretical, 54 (39) 395202, 395202. doi: 10.1088/1751-8121/ac1ee8
N-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra SW(N)
Correa, Francisco, Fazlul Hoque, Md, Marquette, Ian and Zhang, Yao-Zhong (2021). N-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra SW(N). Journal of Physics A: Mathematical and Theoretical, 54 (39) 395201, 395201. doi: 10.1088/1751-8121/ac1dc1
Non-linear ladder operators and coherent states for the 2:1 oscillator
Moran, James, Hussin, Véronique and Marquette, Ian (2021). Non-linear ladder operators and coherent states for the 2:1 oscillator. Journal of Physics A: Mathematical and Theoretical, 54 (27) 275301, 1-17. doi: 10.1088/1751-8121/ac0200
Embedding of the Racah algebra R(n) and superintegrability
Latini, Danilo, Marquette, Ian and Zhang, Yao-Zhong (2021). Embedding of the Racah algebra R(n) and superintegrability. Annals of Physics, 426 168397, 1-18. doi: 10.1016/j.aop.2021.168397
Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems
Campoamor-Stursberg, Rutwig and Marquette, Ian (2021). Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems. Annals of Physics, 424 168378, 168378. doi: 10.1016/j.aop.2020.168378
Zelaya, K., Marquette, I. and Hussin, V. (2020). Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians. Journal of Physics A: Mathematical and Theoretical, 54 (1) 015206, 1-29. doi: 10.1088/1751-8121/abcab8
Polynomial algebras from su(3) and a quadratically superintegrable model on the two sphere
Correa, F., del Olmo, M. A., Marquette, I. and Negro, J. (2020). Polynomial algebras from su(3) and a quadratically superintegrable model on the two sphere. Journal of Physics A: Mathematical and Theoretical, 54 (1) 015205, 1-16. doi: 10.1088/1751-8121/abc909
A fourth-order superintegrable system with arational potential related to Painlevé VI
Marquette, Ian, Post, Sarah and Ritter, Lisa (2020). A fourth-order superintegrable system with arational potential related to Painlevé VI. Journal of Physics A: Mathematical and Theoretical, 53 (50) 50LT01, 1-13. doi: 10.1088/1751-8121/abbf06
A new way to classify 2D higher order quantum superintegrable systems
Berntson, Bjorn Karl, Marquette, Ian and Miller, Willard (2020). A new way to classify 2D higher order quantum superintegrable systems. Journal of Physics A: Mathematical and Theoretical, 53 (49) 494003, 1-21. doi: 10.1088/1751-8121/abc04a
The general Racah algebra as the symmetry algebra of generic systems on pseudo–spheres
Kuru, Sengul, Marquette, Ian and Negro, Javier (2020). The general Racah algebra as the symmetry algebra of generic systems on pseudo–spheres. Journal of Physics A: Mathematical and Theoretical, 53 (40) 405203, 405203. doi: 10.1088/1751-8121/abadb7
Chen, Zhe, Marquette, Ian and Zhang, Yao-Zhong (2019). Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras. Annals of Physics, 411 167970. doi: 10.1016/j.aop.2019.167970
Generalized conformal pseudo-Galilean algebras and their Casimir operators
Campoamor-Stursberg, Rutwig and Marquette, Ian (2019). Generalized conformal pseudo-Galilean algebras and their Casimir operators. Journal of Physics A: Mathematical and Theoretical, 52 (47) 475202, 1-17. doi: 10.1088/1751-8121/ab4c81
Hoffmann, Scott E., Hussin, Véronique, Marquette, Ian and Zhang, Yao-Zhong (2019). Ladder operators and coherent states for multi-step supersymmetric rational extensions of the truncated oscillator. Journal of Mathematical Physics, 60 (5) 052105, 052105. doi: 10.1063/1.5091953
Two-dimensional superintegrable systems from operator algebras in one dimension
Marquette, Ian, Sajedi, Masoumeh and Winternitz, Pavel (2019). Two-dimensional superintegrable systems from operator algebras in one dimension. Journal of Physics A: Mathematical and Theoretical, 52 (11) 115202, 115202. doi: 10.1088/1751-8121/ab01a2
Extended Laplace-Runge-Lentz vectors, new family of superintegrable systems and quadratic algebras
Chen, Zhe, Marquette, Ian and Zhang, Yao-Zhong (2019). Extended Laplace-Runge-Lentz vectors, new family of superintegrable systems and quadratic algebras. Annals of Physics, 402, 78-90. doi: 10.1016/j.aop.2019.01.009
On Casimir operators of conformal Galilei algebras
Alshammari, Fahad, Isaac, Phillip S. and Marquette, Ian (2019). On Casimir operators of conformal Galilei algebras. Journal of Mathematical Physics, 60 (1) 013509, 013509. doi: 10.1063/1.5064840
Extended Calogero models: a construction for exactly solvable kN-body systems
Chen, Zhe, Links, Jon, Marquette, Ian and Zhang, Yao-Zhong (2018). Extended Calogero models: a construction for exactly solvable kN-body systems. Journal of Physics A: Mathematical and Theoretical, 51 (45) 455203, 455203. doi: 10.1088/1751-8121/aae4cc
Coherent states for ladder operators of general order related to exceptional orthogonal polynomials
Hoffmann, Scott E., Hussin, Veronique, Marquette, Ian and Zhang, Yao-Zhong (2018). Coherent states for ladder operators of general order related to exceptional orthogonal polynomials. Journal of Physics A: Mathematical and Theoretical, 51 (31) 315203, 315203. doi: 10.1088/1751-8121/aacb3b
Quantum superintegrable system with a novel chain structure of quadratic algebras
Liao, Yidong, Marquette, Ian and Zhang, Yao-Zhong (2018). Quantum superintegrable system with a novel chain structure of quadratic algebras. Journal of Physics A: Mathematical and Theoretical, 51 (25) 255201. doi: 10.1088/1751-8121/aac111
Recurrence approach and higher order polynomial algebras for superintegrable monopole systems
Hoque, Md Fazlul, Marquette, Ian and Zhang, Yao-Zhong (2018). Recurrence approach and higher order polynomial algebras for superintegrable monopole systems. Journal of Mathematical Physics, 59 (5) 052101, 052101. doi: 10.1063/1.5012859
Hoque, Md. Fazlul, Marquette, Ian, Post, Sarah and Zhang, Yao-Zhong (2018). Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials. Annals of Physics, 391, 203-215. doi: 10.1016/j.aop.2018.02.008
Hoffmann, Scott E., Hussin, Veronique, Marquette, Ian and Zhang, Yao-Zhong (2018). Non-classical behaviour of coherent states for systems constructed using exceptional orthogonal polynomials. Journal of Physics A-Mathematical and Theoretical, 51 (8) 085202, 1-16. doi: 10.1088/1751-8121/aaa553
Alshammari, Fahad, Isaac, Phillip S. and Marquette, Ian (2018). A differential operator realisation approach for constructing Casimir operators of non-semisimple Lie algebras. Journal of Physics A: Mathematical and Theoretical, 51 (6) 065206, 065206. doi: 10.1088/1751-8121/aaa468
Marquette, Ian, Sajedi, Masoumeh and Winternitz, Pavel (2017). Fourth order superintegrable systems separating in Cartesian coordinates I. Exotic quantum potentials. Journal of Physics A: Mathematical and Theoretical, 50 (31) 315201, 315201. doi: 10.1088/1751-8121/aa7a67
Hoque, Md Fazlul, Marquette, Ian and Zhang, Yao-Zhong (2017). Quadratic algebra structure in the 5D Kepler system with non-central potentials and Yang–Coulomb monopole interaction. Annals of Physics, 380, 121-134. doi: 10.1016/j.aop.2017.03.003
Quadratic algebra for superintegrable monopole system in a Taub-NUT space
Hoque, Md Fazlul, Marquette, Ian and Zhang, Yao-Zhong (2016). Quadratic algebra for superintegrable monopole system in a Taub-NUT space. Journal of Mathematical Physics, 57 (9) 092104, 092104. doi: 10.1063/1.4962924
Marquette, Ian and Quesne, Christiane (2016). Connection between quantum systems involving the fourth Painlevé transcendent and k-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. Journal of Mathematical Physics, 57 (5) 052101, 052101. doi: 10.1063/1.4949470
Recurrence approach and higher rank cubic algebras for the N-dimensional superintegrable systems
Hoque, Md Fazlul, Marquette, Ian and Zhang, Yao-Zhong (2016). Recurrence approach and higher rank cubic algebras for the N-dimensional superintegrable systems. Journal of Physics A: Mathematical and Theoretical, 49 (12) 125201, 1-12. doi: 10.1088/1751-8113/49/12/125201
Isaac, Phillip S. and Marquette, Ian (2016). Families of 2D superintegrable anisotropic Dunkl oscillators and algebraic derivation of their spectrum. Journal of Physics A: Mathematical and Theoretical, 49 (11) 115201, 1-13. doi: 10.1088/1751-8113/49/11/115201
Hoque, Md Fazlul, Marquette, Ian and Zhang, Yao-Zhong (2015). A new family of N dimensional superintegrable double singular oscillators and quadratic algebra Q(3) ⊕ so(n) ⊕ so(N-n). Journal of Physics A: Mathematical and Theoretical, 48 (Art No.: 445207) 445207, 273-&. doi: 10.1088/1751-8113/48/44/445207
Exact solution of the p+IP Hamiltonian revisited: Duality relations in the hole-pair picture
Links, Jon, Marquette, Ian and Moghaddam, Amir (2015). Exact solution of the p+IP Hamiltonian revisited: Duality relations in the hole-pair picture. Journal of Physics A: Mathematical and Theoretical, 48 (37) 374001, 1-22. doi: 10.1088/1751-8113/48/37/374001
Bagchi, Bijan and Marquette, Ian (2015). New 1-step extension of the Swanson oscillator and superintegrability of its two-dimensional generalization. Physics Letters, Section A: General, Atomic and Solid State Physics, 379 (26-27), 1584-1588. doi: 10.1016/j.physleta.2015.04.009
Marquette, Ian and Quesne,Christine (2015). Deformed oscillator algebra approach of some quantum superintegrable Lissajous systems on the sphere and of their rational extensions. Journal of Mathematical Physics, 56 (6) 062102, 062102-1-062102-19. doi: 10.1063/1.4922020
Quadratic algebra structure and spectrum of a new superintegrable system in N-dimension
Hoque, Md Fazlul, Marquette, Ian and Zhang, Yao-Zhong (2015). Quadratic algebra structure and spectrum of a new superintegrable system in N-dimension. Journal of Physics A: Mathematical and Theoretical, 48 (18) 185201, 1-16. doi: 10.1088/1751-8113/48/18/185201
Ground-state Bethe root densities and quantum phase transitions
Links, Jon and Marquette, Ian (2015). Ground-state Bethe root densities and quantum phase transitions. Journal of Physics A: Mathematical and Theoretical, 48 (4) 045204, 1-15. doi: 10.1088/1751-8113/48/4/045204
Marquette, Ian and Quesne, Christiane (2014). Combined state-adding and state-deleting approaches to type III multi-step rationally extended potentials: applications to ladder operators and superintegrability. Journal of Mathematical Physics, 55 (11) 112103, 112103-1-112103-25. doi: 10.1063/1.4901006
New quasi-exactly solvable class of generalized isotonic oscillators
Agboola, Davids, Links, Jon, Marquette, Ian and Zhang, Yao-Zhong (2014). New quasi-exactly solvable class of generalized isotonic oscillators. Journal of Physics A: Mathematical and Theoretical, 47 (39) 395305, 395305.1-395305.17. doi: 10.1088/1751-8113/47/39/395305
On realizations of polynomial algebras with three generators via deformed oscillator algebras
Isaac, Phillip S. and Marquette, Ian (2014). On realizations of polynomial algebras with three generators via deformed oscillator algebras. Journal of Physics A: Mathematical and Theoretical, 47 (20) 205203, 1-26. doi: 10.1088/1751-8113/47/20/205203
Marquette, Ian and Quesne, Christiane (2013). New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. Journal of Mathematical Physics, 54 (10) 102102, 102102-1-102102-12. doi: 10.1063/1.4823771
Marquette, Ian (2013). Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras. Journal of Mathematical Physics, 54 (7) 071702, 071702.1-071702.15. doi: 10.1063/1.4816086
Marquette, I. and Quesne, C. (2013). Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators. Journal of Physics A: Mathematical and Theoretical, 46 (15) 155201, 155201. doi: 10.1088/1751-8113/46/15/155201
New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials
Marquette, Ian and Quesne, Christiane (2013). New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials. Journal of Mathematical Physics, 54 (4) 042102, 042102-1-042102-16. doi: 10.1063/1.4798807
Integrability of an extended d+id-wave pairing Hamiltonian
Marquette, Ian and Links, Jon (2013). Integrability of an extended d+id-wave pairing Hamiltonian. Nuclear Physics B, 866 (3), 378-390. doi: 10.1016/j.nuclphysb.2012.09.006
Singular isotonic oscillator, supersymmetry and superintegrability
Marquette, Ian (2012). Singular isotonic oscillator, supersymmetry and superintegrability. Symmetry Integrability and Geometry: Methods and Applications, 8 063. doi: 10.3842/SIGMA.2012.063
Marquette, Ian and Links, Jon (2012). Generalized Heine-Stieltjes and Van Vleck polynomials associated with two-level, integrable BCS models. Journal of Statistical Mechanics: Theory and Experiment, 2012 (8) P08019, P08019. doi: 10.1088/1742-5468/2012/08/P08019
Generalized five-dimensional Kepler system, Yang-Coulomb monopole, and Hurwitz transformation
Marquette, Ian (2012). Generalized five-dimensional Kepler system, Yang-Coulomb monopole, and Hurwitz transformation. Journal of Mathematical Physics, 53 (2) 022103, 022103.1-022103.12. doi: 10.1063/1.3684955
Marquette, Ian (2012). Classical ladder operators, polynomial Poisson algebras, and classification of superintegrable systems. Journal of Mathematical Physics, 53 (1) 012901, 012901.1-012901.12. doi: 10.1063/1.3676075
Generalized Kaluza-Klein monopole, quadratic algebras and ladder operators
Marquette, Ian (2011). Generalized Kaluza-Klein monopole, quadratic algebras and ladder operators. Journal of Physics A-Mathematical and Theoretical, 44 (23) 235203, 235203.1-235203.12. doi: 10.1088/1751-8113/44/23/235203
Quadratic algebra approach to relativistic quantum Smorodinsky-Winternitz systems
Marquette, Ian (2011). Quadratic algebra approach to relativistic quantum Smorodinsky-Winternitz systems. Journal of Mathematical Physics, 52 (4) 042301, 042301-1-042301-12. doi: 10.1063/1.3579983
Generalized MICZ-Kepler system, duality, polynomial, and deformed oscillator algebras
Marquette, Ian (2010). Generalized MICZ-Kepler system, duality, polynomial, and deformed oscillator algebras. Journal of Mathematical Physics, 51 (10) 102105, 102105-1-102105-10. doi: 10.1063/1.3496900
Marquette, Ian (2010). Construction of classical superintegrable systems with higher order integrals of motion from ladder operators. Journal of Mathematical Physics, 51 (7) 037006JMP, 072903-1-072903-9. doi: 10.1063/1.3448925
Superintegrability and higher order polynomial algebras
Marquette, Ian (2010). Superintegrability and higher order polynomial algebras. Journal of Physics A - Mathematical and Theoretical, 43 (13) 135203, 135203-1-135203-15. doi: 10.1088/1751-8113/43/13/135203
Marquette, Ian (2009). Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion. Journal of Mathematical Physics, 50 (12) 122102, 122102-1-122102-10. doi: 10.1063/1.3272003
Marquette, Ian (2009). Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II. Painleve transcendent potentials. Journal of Mathematical Physics, 50 (9), 095202-1-095202-18. doi: 10.1063/1.3096708
Marquette, Ian (2009). Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials. Journal of Mathematical Physics, 50 (1), 012101-1-012101-23. doi: 10.1063/1.3013804
Superintegrable systems with third-order integrals of motion
Marquette, Ian and Winternitz, Pavel (2008). Superintegrable systems with third-order integrals of motion. Journal of Physics A - Mathematical and Theoretical, 41 (30) 304031, 304031-1-304031-10. doi: 10.1088/1751-8113/41/30/304031
Marquette, I. and Winternitz, P. (2008). Erratum : Polynomial poisson algebras for classical superintegrable systems with a third order integral of motion (vol 48, art no 012902, 2007). Journal of Mathematical Physics, 49 (1) 019901, 019901. doi: 10.1063/1.2831929
Polynomial Poisson algebras for superintegrable systems with a third order integral of motion
Marquette, Ian and Winternitz, Pavel (2007). Polynomial Poisson algebras for superintegrable systems with a third order integral of motion. Journal of Mathematical Physics, 48 (1), 012902.1-012902.16. doi: 10.1063/1.2399359
A New Approach to Analysis of 2D Higher Order Quantum Superintegrable Systems
Berntson, Bjorn K., Marquette, Ian and Miller, Willard (2020). A New Approach to Analysis of 2D Higher Order Quantum Superintegrable Systems. 11th International Symposium on Quantum Theory and Symmetries, Montreal, Canada, 1-5 July 2019. Cham, Switzerland: Springer. doi: 10.1007/978-3-030-55777-5_10
Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras
Marquette, Ian (2019). Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras. 32nd International Colloquium on Group Theoretical Methods in Physics, ICGTMP 2018, Prague, Czech Republic, 9-13 July, 2018. Bristol, United Kingdom: Institute of Physics Publishing. doi: 10.1088/1742-6596/1194/1/012074
Hoffmann, Scott E., Hussin, Véronique, Marquette, Ian and Zhang, Yao-Zhong (2019). Coherent states for rational extensions and ladder operators related to infinite-dimensional representations. XXVI International Conference on Integrable Systems and Quantum symmetries, Prague, Czech Republic, 8–12 July 2019. Bristol, United Kingdom: Institute of Physics Publishing. doi: 10.1088/1742-6596/1416/1/012013
On superintegrable monopole systems
Hoque, Md Fazlul, Marquette, Ian and Zhang, Yao-Zhong (2018). On superintegrable monopole systems. 25th International Conference on Integrable Systems and Quantum Symmetries, ISQS 2017, Prague, Czech Republic, 6-10June 2017. Bristol, United Kingdom: Institute of Physics Publishing. doi: 10.1088/1742-6596/965/1/012018
Family of N-dimensional superintegrable systems and quadratic algebra structures
Hoque, Md Fazlul, Marquette, Ian and Zhang, Yao-Zhong (2016). Family of N-dimensional superintegrable systems and quadratic algebra structures. 23rd International Conference on Integrable Systems and Quantum Symmetries, ISQS 2015, Prague, Czech Republic, 23 - 27 June 2015. Bristol, United Kingdom: Institute of Physics Publishing. doi: 10.1088/1742-6596/670/1/012024
Marquette, Ian (2015). New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies. 30th International Colloquium on Group Theoretical Methods in Physics (Group30), Ghent, Belgium, 14-18 July 2014. Bristol, United Kingdom: Institute of Physics Publishing. doi: 10.1088/1742-6596/597/1/012057
An infinite family of superintegrable systems from higher order ladder operators and supersymmetry
Marquette, Ian (2011). An infinite family of superintegrable systems from higher order ladder operators and supersymmetry. GROUP 28 Conference: XXVIIIth International Colloquium on Group-Theoretical Methods in Physics (ICGTMP), Newcastle upon Tyne, United Kingdom, 26–30 July 2010. Bristol, United Kingdom: Institute of Physics Publishing. doi: 10.1088/1742-6596/284/1/012047
Generalized Heisenberg algebras, SUSYQM and degeneracies: Infinite well and Morse potential
Hussin, Veronique and Marquette, Ian (2011). Generalized Heisenberg algebras, SUSYQM and degeneracies: Infinite well and Morse potential. Workshop on Supersymmetric Quantum Mechanics and Spectral Design, Benasque, Spain, 18-30 July 2010. Kyiv, Ukraine: Natsional'na Akademiya Nauk Ukrainy. doi: 10.3842/SIGMA.2011.024
Marquette, Ian (2008). Polynomial associative algebras for quantum superintegrable systems with a third order integral of motion. Institute for Mathematics and its Applications: Summer Program 2006, Minneapolis, MN, U.S.A., 17 July-4 August, 2006. New York, NY, United States: Springer New York. doi: 10.1007/978-0-387-73831-4_24
Representation theory in exactly solvable systems
(2019–2024) ARC Future Fellowships
New constructions of superintegrable systems and the connection with Painleve transcendents
(2013–2016) ARC Discovery Early Career Researcher Award
ResTeach Funding 2012 0.1 FTE School of Math & Physics
(2012–2013) UQ ResTeach
Quadratic Algebras and Their Casimir Invariant
Doctor Philosophy — Principal Advisor
Other advisors:
Superintegrable systems with Dunkl operators, their symmetry algebras and invariant
Doctor Philosophy — Principal Advisor
Other advisors:
Superintegrabe systems in curved spaces and algebraic approaches
Doctor Philosophy — Associate Advisor
Other advisors:
Deformations of supersymmetric quantum field theories and gravity
Doctor Philosophy — Associate Advisor
Coherent states and scattering for exactly solvable quantum systems
(2020) Doctor Philosophy — Associate Advisor
Other advisors:
New families of exactly solvable many-body models and superintegrable systems
(2019) Master Philosophy — Associate Advisor
Other advisors:
On the construction of Casimir operators for non-semisimple Lie algebras
(2018) Doctor Philosophy — Associate Advisor
Other advisors:
Superintegrable systems, polynomial algebra structures and exact derivations of spectra
(2018) Doctor Philosophy — Associate Advisor
Other advisors:
Note for students: The possible research projects listed on this page may not be comprehensive or up to date. Always feel free to contact the staff for more information, and also with your own research ideas.
Construction of Casimir operators of higher rank quadratic algebras
This project will intend to make new discoveries in regard of quadratic algegbras which naturally occur in context of integrable systems and to develop method to develop to obtain their Casimir operators. These results will allow to make further progress in their representation theory and obtain the spectrum of new superintegrable systems.
Indecomposable representations and realizations related to exactly solvable systems
This project will intend to develop differential operator realizations connected to indecomposable representations and apply to the construction of exactly solvable systems.