# Bispectral Problem and Quantum Integrable Systems (Integrable Systems)

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• Dr O Chalykh
• Applications accepted all year round
• Competition Funded PhD Project (European/UK Students Only)

## Project Description

Some ordinary differential operators have the following curious property: their eigenfunctions satisfy differential equations in the spectral variable. The simplest example is L=d/dx: its eigenfunctions f(x)=exp(kx) satisfy Lf=kf, but at the same time f (as a function of k) satisfies df/dk=xf. This property is called 'bispectrality'. In the previous example it is a trivial corollary of the symmetry f(k,x)=f(x,k) between the 'spatial' variable x and the 'spectral' variable k. However, in other cases this kind of symmetry can be much more sophisticated and far less obvious. Bispectral problem asks for a classification of such operators L. This question first appeared in relation to some problems in tomography. Surprisingly, this problem and its multivariable version have close links to integrable nonlinear PDEs, such as the Korteweg-de Vries (KdV) equation, as well as to quantum integrable systems of Calogero-Moser type. There are many algebraic aspects of bispectrality, both in dimension one and in higher dimension, which need better understanding and deserve further study. Replacing differential operators by difference ones allows further generalizations with the links to the theory of mutivariable orthogonal polynomials and special functions.

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 six permanent members of staff work with five 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.