# Dr Ian Marquette

## Researcher biography

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

### Book Chapter

*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

### Journal Articles

*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

*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*. Journal of Mathematical Physics, 63 (12) 121703, 1-23. doi: 10.1063/5.0096769

*Dynamical symmetry algebras of two superintegrable two-dimensional systems*. Journal of Physics A: Mathematical and Theoretical, 55 (41) 415203, 415203. doi: 10.1088/1751-8121/ac9164

*Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators*. Physica D: Nonlinear Phenomena, 440 133464, 133464. doi: 10.1016/j.physd.2022.133464

*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*. Journal of Physics A: Mathematical and Theoretical, 55 (15) 155201. doi: 10.1088/1751-8121/ac550a

*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

*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

*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*. 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*. 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)*. 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*. 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*. 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*. Annals of Physics, 424 168378, 168378. doi: 10.1016/j.aop.2020.168378

*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*. 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*. 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*. 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*. Journal of Physics A: Mathematical and Theoretical, 53 (40) 405203, 405203. doi: 10.1088/1751-8121/abadb7

*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*. Journal of Physics A: Mathematical and Theoretical, 52 (47) 475202, 1-17. doi: 10.1088/1751-8121/ab4c81

*Ladder operators and coherent states for multi-step supersymmetric rational extensions of the truncated oscillator*. Journal of Mathematical Physics, 60 (5) 052105. doi: 10.1063/1.5091953

*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*. Annals of Physics, 402, 78-90. doi: 10.1016/j.aop.2019.01.009

*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*. 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*. 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*. 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*. Journal of Mathematical Physics, 59 (5) 052101, 052101. doi: 10.1063/1.5012859

*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

*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

*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

*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

*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*. Journal of Mathematical Physics, 57 (9) 092104, 092104. doi: 10.1063/1.4962924

*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*. Journal of Physics A: Mathematical and Theoretical, 49 (12) 125201, 1-12. doi: 10.1088/1751-8113/49/12/125201

*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

*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*. Journal of Physics A: Mathematical and Theoretical, 48 (37) 374001, 1-22. doi: 10.1088/1751-8113/48/37/374001

*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

*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*. 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*. Journal of Physics A: Mathematical and Theoretical, 48 (4) 045204, 1-15. doi: 10.1088/1751-8113/48/4/045204

*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*. 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*. Journal of Physics A: Mathematical and Theoretical, 47 (20) 205203, 1-26. doi: 10.1088/1751-8113/47/20/205203

*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

*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

*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*. 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*. Nuclear Physics B, 866 (3), 378-390. doi: 10.1016/j.nuclphysb.2012.09.006

*Singular isotonic oscillator, supersymmetry and superintegrability*. Symmetry Integrability and Geometry: Methods and Applications, 8 063. doi: 10.3842/SIGMA.2012.063

*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*. Journal of Mathematical Physics, 53 (2) 022103, 022103.1-022103.12. doi: 10.1063/1.3684955

*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*. 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*. 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*. Journal of Mathematical Physics, 51 (10) 102105, 102105-1-102105-10. doi: 10.1063/1.3496900

*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*. Journal of Physics A - Mathematical and Theoretical, 43 (13), 135203-1-135203-15. doi: 10.1088/1751-8113/43/13/135203

*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

*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

*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*. Journal of Physics A - Mathematical and Theoretical, 41 (30), 304031-1-304031-10. doi: 10.1088/1751-8113/41/30/304031

*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. doi: 10.1063/1.2831929

*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

### Conference Papers

*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*.

*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

*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*.

*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*.

*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

*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*.

*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*.

*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

*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