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Abstract

The Lanczos algorithm is applied to solve relativistic quantum mechanical problems based on the Dirac and Klein-Gordon equation numerically. This method is applicable when the action of the Hamiltonian on a state vector is known. No further requirements on the Hamiltonian, or the underlying discretization, are necessary. It is demonstrated, that the Lanczos algorithm can be used to precisely calculate approximate bound states of relativistic Hamiltonians, as well as their energies. Our results show that the Lanczos algorithm converges very fast to bound states of relativistic Hamiltonians, despite their energy spectrum not being bounded. It is also shown that very precise numerical time propagation with arbitrary electromagnetic fields can be performed. This is demonstrated in one and two dimensions, the precision is analyzed by means of free wave packets where an analytical solution can be computed. The application of the Lanczos algorithm is especially attractive because it only requires matrix-vector and vector products, which can easily be parallelized. This makes the Lanczos algorithm an ideal tool for large scale parallel calculations. We demonstrate the excellent parallelization capabilities of the Lanczos algorithm by an implementation of the relativistic time-evolution operator based on the Message Passing Interface standard.