Scattering resonances and the complex absorbing potential method

Scattering resonances are the analogues of eigenvalues for problems on non-compact domains. The real part and imaginary part of the resonances capture the rates of oscillation and decay of the scattering waves. Hence the location of resonances reflects the long-time behavior of the waves on non-compact domains.In this thesis we study a computational technique for scattering resonances, that is the method of \emph{complex absorbing potentials} (CAP). We show that the CAP method for computing resonances applies to the case of scattering by exponentially decaying potentials. We also show that the CAP method is valid for an abstractly defined class of \emph{black box} perturbations of dilation analytic second order differential operators which is close to the Laplacian near infinity. The black box formalism allows a unifyingtreatment of diverse problems ranging from obstacle scattering to scattering on finite volume surfaces without addressing the details of specific situations.The black box scattering problem motivates us to study the boundary perturbations in obstacle scattering. We show that all resonances in obstacle scattering with Dirichlet boundary condition are generically simple in the class of obstacles with $C^k$ (and $C^\infty$) boundaries, $k \geq 2$. This generalizes the case of eigenvalues of second order elliptic operators on a compact domain that all eigenvalues are simple for a generic compact domain