Jacobi-Davidson Algorithm for Locating Resonances in a Few-Body Tunneling System

Abstract

A recent theoretical study of quantum few-body tunneling implemented a model using a Berggren basis expansion. This approach leads to eigenvalue problems, involving large, complex-symmetric Hamiltonian matrices. In addition, the eigenspectrum consists mainly of irrelevant scattering states. The physical resonance is usually hidden somewhere in the continuum of these scattering states, making diagonalization difficult. This thesis describes the theory of the Jacobi-Davidson algorithm for calculating complex eigenvalues and thus identifying the resonance energies of interest. The underlying Davidson method is described and combined with Jacobi's orthogonal complement method to form the Jacobi-Davidson algorithm. The algorithm is implemented and applied to matrices from the theoretical study. Furthermore, a non-hermitian formulation of quantum mechanics is introduced and the Berggren basis expansion explained. The results show that the ability of the Jacobi-Davidson algorithm to locate a specific interior eigenvalue greatly reduces the computational times compared to previous diagonalization methods. However, the additional computational cost of implementing the Jacobi correction turns out to be unnecessary in this application; thus, the Davidson algorithm is sufficient for finding the resonance state of these matrices.