报告题目 Title: Painlevé transcendents in quantum mechanics and algebraic aspects

报告人 Speaker: Ian Marquette

报告人所在单位 Affiliation: La Trobe University

时间 Time: 2025-12-29 13:30-14:30

地点 Venue: Room 1601, Guanghua Eastern Main Tower (Handan Campus)

报告摘要 Abstract: I will discuss the six Painlevé transcendents that were obtained by Painlevé, Gambier, and Fuchs in the early 1900s. They play an important role in the classification of ordinary differential equations, and they are related to several areas in physics and mathematics, such as the reduction of various nonlinear differential equations of mathematical physics, relativity, statistical mechanics, and quantum field theory. Their connection with non-relativistic quantum mechanics is much more recent. The Painlevé transcendents appear  in context of the potential of quantum Hamiltonians obtained via higher order Darboux transformations, operator algebras and superintegrability.

I will do an overview of quantum systems involving the Painlevé transcendents, their known properties and related algebraic structures. This includes one-dimensional Hamiltonian and two-dimensional systems on Euclidean and Riemannian spaces. I will highlight results for Hamiltonians whose potentials are written in terms of the fourth and sixth Painlevé transcendents. I will also discuss the connection with exceptional orthogonal polynomials of Hermite, Laguerre and Jacobi type. The model involving the sixth Painlevé transcendent has a cubic algebra, which is exploited to obtain the energy spectrum via finite dimensional representations.

个人简介 Bio: Dr. Ian Marquette obtained a Ph.D. in Physics from the Université de Montréal in 2009. Following this, he was awarded an FQRNT fellowship for the project Integrability, Superintegrability, Algebraic Structures and Applications, which took him to the University of York in the United Kingdom, where he spent two years in the Department of Mathematics. In 2011, Dr. Marquette moved to Australia to take up a postdoctoral position at The University of Queensland. There, he focused on exactly solvable and integrable models related to superconductivity and phase transitions, as well as on indecomposable representations of Lie algebras.

His work has been recognised through several prestigious awards, including an ARC Discovery Early Career Researcher Award from 2013 to 2016 for the project New Constructions of Superintegrable Systems and the Connection with Painlevé Transcendents, and an ARC Future Fellowship from 2019 to 2024 for the project Representation Theory in Exactly Solvable Systems. Throughout his career, Dr. Marquette has contributed broadly to mathematical physics, with a particular emphasis on quantum models, their symmetries and related special functions.

Since January 2024, Dr. Marquette has been a Senior Lecturer in the Department of Mathematics and Physical Sciences at La Trobe University, where he continues to develop new directions in mathematical physics and explore connections with other areas of mathematics.

海报 Poster: Ian Marquette 学术报告.jpg