Resonant state expansion for relativistic and non-relativistic wave equations (physics)

The resonant state expansion (RSE) is a rigorous perturbative method recently invented in Cardiff School of Physics and Astronomy. The RSE has been developed and used for relativistic wave equations describing such open systems as planar, cylindrical and spherical optical resonators with various perturbations. The method was demonstrated to be particularly suitable for calculating highquality
modes in such open system and efficient and accurate in calculating their perturbations. It has been shown that the RSE is a few orders of magnitude quicker computational tool as compared to existing commercial software packages like ComSol or Lumerical.
One of the main aims of the project is to apply the RSE to non-relativistic wave equations and develop a theory of resonant states in quantum-mechanical systems. In particular, the RSE will be applied to Schrödinger’s wave equation describing an electron/exciton in various nanostructures with effectively one-, two- and/or three-dimensional potentials and various perturbations. The efficiency and the quality of the new method will be investigated and tested against the existing finite difference methods and ab initio calculations. The concept of resonant states will also be used to investigate the short and long-time behaviour of quantum states and to calculate a non-quadratic (non-exponential) decay of the survival probability of quantum states at short (long) times.
The project will also consider some further applications of the RSE in relativistic problems. This may include calculation of eigenstates in optical planar waveguides and photonic crystal fibres where everlasting waveguide modes contribute to expansions of the Green’s function. Finally, there will be made attempts towards generalisation of the RSE for relativistic Dirac wave equation with a
potential application of the method to narrow-band semiconductor nanostructures.