My project involves finding solutions to the (time independent) Shroedinger equation on perturbed regions of a sphere or ball in order to find resonant states for decay rates within that region.
As resonant states in such a region behave much like bound states, an interpolation is attempted between numerical solutions within this region with an added boundary condition imposed on the adjacent part of the ball's circumference, and analytical solutions for the region outside sphere. The boundary condition is smoothed by successive refinements until internal and external eigenfunctions 'match.' Then, these are used to calculate the decay rate. From this set of solutions, resonance states are exactly those with a low decay rate.