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: 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: . doi:10.1088/1751-8121/aacb3b
Journal Article: 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: . doi:10.1088/1751-8121/aac111
Journal Article: 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: . doi:10.1063/1.5012859
Representation theory in exactly solvable systems
(2018–2022) ARC Future Fellowships
New constructions of superintegrable systems and the connection with Painleve transcendents
(2013–2016) ARC Discovery Early Career Researcher Award
Aspects of exactly solvable and superintegrable models involving exceptional orthogonal polynomials
Doctor Philosophy
On the construction of Casimir operators for non-semisimple Lie algebras
Doctor Philosophy
Aspects of quasi-exactly solvable systems in quantum mechanics
Master 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.
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: . 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: . 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: . 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: 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: . 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: . 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: . 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: . 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: 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: 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: . 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: 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: 062102-1-062102-19. 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: 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: 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: 112103-1-112103-25. 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: 395305.1-395305.17. 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: 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: 102102-1-102102-12. 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: 071702.1-071702.15. 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: . 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: 042102-1-042102-16. 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 . 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: . 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: 022103.1-022103.12. 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: 012901.1-012901.12. 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: 235203.1-235203.12. 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: 042301-1-042301-12. 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: 102105-1-102105-10. 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: 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: 135203-1-135203-15. 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: 122102-1-122102-10. 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: 304031-1-304031-10. 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: . 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
On superintegrable monopole systems
Hoque, Md Fazlul, Marquette, Ian and Zhang, Yao-Zhong (2018). On superintegrable monopole systems. In: Journal of Physics: Conference Series. 25th International Conference on Integrable Systems and Quantum Symmetries, ISQS 2017, Prague, Czech Republic, (1-8). June 6-10, 2017. 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. In: C. Burdik, O. Navratil and S. Posta, XXIII International Conference on Integrable Systems and Quantum Symmetries (ISQS-23). 23rd International Conference on Integrable Systems and Quantum Symmetries, ISQS 2015, Prague, Czech Republic, (). 23 - 27 June 2015. 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. In: 30th International Colloquium on Group Theoretical Methods in Physics (Group30), Ghent, Belgium, (012057.1-012057.10). 14-18 July 2014. 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. In: GROUP 28: Physical and mathematical aspects of symmetry: Proceedings of the 28th International Colloquium on Group-Theoretical Methods in Physics. GROUP 28 Conference: XXVIIIth International Colloquium on Group-Theoretical Methods in Physics (ICGTMP), Newcastle upon Tyne, United Kingdom, (012047-1-012047-8). 26–30 July 2010. 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. In: Alexander Andrianov, Veronique Hussin, Javier Negro, Luismi Nieto and Andrei Smilga, Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”. Workshop on Supersymmetric Quantum Mechanics and Spectral Design, Benasque, Spain, (024.1-024.16). 18-30 July 2010. doi:10.3842/SIGMA.2011.024
Marquette, Ian (2008). Polynomial associative algebras for quantum superintegrable systems with a third order integral of motion. In: Michael Eastwood and Willard Miller, Symmetries and Overdetermined Systems of Partial Differential Equations. Proceedings. Institute for Mathematics and its Applications: Summer Program 2006, Minneapolis, MN, U.S.A., (461-469). 17 July-4 August, 2006. doi:10.1007/978-0-387-73831-4_24
Representation theory in exactly solvable systems
(2018–2022) ARC Future Fellowships
New constructions of superintegrable systems and the connection with Painleve transcendents
(2013–2016) ARC Discovery Early Career Researcher Award
Aspects of exactly solvable and superintegrable models involving exceptional orthogonal polynomials
Doctor Philosophy — Associate Advisor
Other advisors:
On the construction of Casimir operators for non-semisimple Lie algebras
Doctor Philosophy — Associate Advisor
Other advisors:
Aspects of quasi-exactly solvable systems in quantum mechanics
Master 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.