Numerical Continuation of resonances and bound states in coupled channel Schrödinger equations

Abstract: In this contribution, we introduce numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, to the problem of tracing the parameter dependence of bound and resonant states of the quantum mechanical Schr\"odinger equation. We extend previous work on the subject to systems of coupled equations. Bound and resonant states of the Schr\"odinger equation can be determined through the poles of the S-matrix, a quantity that can be derived from the asymptotic form of the wave function. We introduce a regularization procedure that essentially transforms the S-matrix into its inverse and improves its smoothness properties, thus making it amenable to numerical continuation. This allows us to automate the process of tracking bound and resonant states when parameters in the Schr\"odinger equation are varied. We have applied this approach to a number of model problems with satisfying results.