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Exact energy-levels of Quantum Lattice problems

Quantum lattices are lattices of "sites" in one or more dimensions which contain a number of quantum particles. The particles can hop from site to site, and various quantum Hamiltonians are used to model the energy states of different systems. There are applications to energy transport on biological molecules, High Temperature Super-Conductivity, theories of quantum dots, and quantum computing. The project involves using computer algebra package to investigate new models by deriving exact representations of the corresponding Hamiltonian matrix, and to investigate the properties of these solutions both numerically and analytically. See the paper (Some exact results for quantum lattice problem) in pdf form. This work is done in collaboration with researchers at the University of Salerno in Italy.

The dynamics of Quantum particles on Classical Nonlinear Lattices

This project looks at a variation of the above theme, where a quantum particle is coupled to a nonlinear classical lattice. The atoms on the lattice are coupled by nonlinear forces, so the lattice vibrates in a complicated way, with localized modes of vibrations. These modes are coupled to quantum particles such as electrons. There are applications to energy transport on biological molecules and High Temperature Super-Conductivity. This work is done in collaboration with researchers at the University of the Algarve in Portugal. See the paper (Dynamical two electron states in a Hubbard-Davydov model) in pdf form.

Hyperelliptic functions and exact solutions to nonlinear wave equations

Many students will have come across elliptic functions like sn and cn, which are periodic versions of the more familiar cosh and sinh functions. These functions are intimately connected with mathematical curves given by equations of the sort y^2=a+bx+cx^2+dx^3. There is a wealth of nice theory associated with elliptic functions, mostly developed in the 19th century, there are many applications, such as the shape of a travelling waves in water, and the motion of a nonlinear pendulum. What is new in this project is extending all this theory to so-called hyperelliptic functions, associated with curves like that above, but with the cubic polynomial in x generalised to a quintic or higher-order polynomials. Although some good work in this area dates back to the start of the last century, much new stuff has been done in the last ten years, aided by powerful computer algebra programs such as Maple. There are many problems still to be studied. An interest in computer algebra, nonlinear waves, and/or algebraic geometry would be useful. This is work in collaboration with researchers in Kiev and Boston.