Edinburgh

# Classical and Quantum Integrability (Integrable Systems)

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• Prof A Fordy
• Applications accepted all year round
• Competition Funded PhD Project (European/UK Students Only)

## Project Description

Classical and Quantum Integrability (Integrable Systems)

According to Liouville's Theorem in classical mechanics, a system of n degrees of freedom and with n Poisson commuting functions (with some technical conditions) is completely integrable. By analogy, a quantum integrable system in n dimensions is one with n commuting differential operators.

A super-integrable system is one for which there are more than n first integrals, with a subset of n Poisson commuting functions. A famous example is the Kepler problem, which has "hidden symmetries" related to the Runge-Lenz vector. Remarkably, such systems can be solved by purely algebraic means, with solving differential equations! These additional integrals form an enlarged Poisson algebra and these are themselves very interesting algebraic structures.

As well as classifying algebras of Poisson commuting functions of various specific classes or algebras of differential operators (both commuting and of ladder type), we can build the spectrum and eigenfunctions of deformations of Laplace-Beltrami operators (with vector and/or scalar potentials) on various differential manifolds.

The subject is closely related to the theory of orthogonal polynomials and other special functions in n-dimensions.

Integrable Systems

The Integrable Systems group in Leeds, one of the leading groups worldwide working on integrable systems, has been strengthened by three recent appointments, one at a senior level. Integrable Systems are systems that, albeit highly nontrivial and nonlinear, are amenable to exact and rigorous techniques for their solvability.

They can take many shapes or forms: nonlinear evolution equations, partial and ordinary differential equations and difference equations, Hamiltonian many-body systems, quantum systems and spin models in statistical mechanics. A large number of mathematical techniques have been developed to unravel the rich structures behind these systems. The five permanent members of staff work with nine postdocs and postgraduate students.

Leeds is a major centre for the theory of discrete and quantum integrable systems.

This group represents a wide range of research activities into integrable nonlinear systems, their symmetries, solution techniques and the underlying mathematical structures, as well as more mathematical aspects of physical systems, for example quantum systems. The models comprise ordinary and partial differential and difference equations, dynamical mappings, discrete Painleve equations, Hamiltonian and many-particle systems and systems of hydrodynamic type. The theory and its specific models have wide-ranging applications, for example, in nonlinear optics, theory of water waves, integrable quantum field theory, statistical mechanics and combinatorics random matrix theory and nanotechnology.

Keywords: Hamiltonian systems; complete integrability; Poisson algebras; ladder operators;