Localization, resonance and non-Hermitian quantum mechanics

A type of non-Hermitian generalization of quantum mechanics is discussed. We introduce an imaginary vector potential to the Hamiltonian of systems such as Anderson-localization systems and mesoscopic systems. In these systems the imaginary part of the wave number of the eigenfunction asymptote ψ(x) eikx is of fundamental interest. We propose the use of the imaginary vector potential for the measurement of the localization length and the resonance lifetime.