Eigenstates and properties of random systems in one dimension at zero temperature

Eigenenergies and eigenfunctions are evaluated. Different localized eigenstates are proven to have exponentially different mobilities. This leads to exponentially high and exponentially narrow resonances of the conductance at eigenenergies. The probability distribution of resonance resistances is determined. The height and the width of a resonance allow one to evaluate the localization position and the localization length of an eigenstate. The phase correlation length is proven to be $\frac{1}{2}{L}_{0}$, where ${L}_{0}$ is the wave-function localization length. The latter strongly depends on the analytical nature of disorder. The dependence of ${L}_{0}$ on energy $\mathcal{E}$ may vary from algebraic to exponential. In the latter case a weak localization is achieved at reasonable energies $\mathcal{E}\ensuremath{\propto}{[\mathrm{ln}(\frac{L}{{L}_{0}})]}^{2}$, where $L$ is the length of the system. Different one-dimensional problems are reduced to the Schr\"odinger equation. In particular, at low frequencies $\ensuremath{\omega}$, acoustic phonons and electromagnetic waves in a random media are localized. Their localization length is ${L}_{0}\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}2}$.