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In this work, we discuss the resonance states of a quantum particle in a periodic potential plus static force. Originally this problem was formulated for a crystalline electron subject to the static electric field and is known nowadays as the Wannier-Stark problem. We describe a novel approach to the Wannier-Stark problem developed in recent years. This approach allows to compute the complex energy spectrum of a Wannier-Stark system as the poles of a rigorously constructed scattering matrix and, in this sense, solves the Wannier-Stark problem without any approximation. The suggested method is very efficient from the numerical point of view and has proven to be a powerful analytic tool for Wannier-Stark resonances appearing in different physical systems like optical or semiconductor superlattices.
An extremely simple and convenient method is presented for computing eigenvalues in quantum mechanics by representing position and momentum operators in a simple matrix form. The simplicity and success of the method is illustrated by numerical results concerning eigenvalues of bound systems and resonances for hermitian and non-hermitian Hamiltonians as well as driven quantum systems.