Publication | Open Access
Localization of Eigenstates and Transport Phenomena in the One-Dimensional Disordered System
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1973
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Spectral TheoryEngineeringRiemann-hilbert ProblemPhysicsPotential TheoryOne-dimensional Disordered SystemApplied PhysicsDisordered Quantum SystemHeat ConductionTransport PhenomenaAnomalous DiffusionQuantum ChaosExponential GrowthIntegrable SystemQuantum Diffusion ProblemMany-body Localization
The study unifies theories of localization, transmission, heat conduction in harmonic chains, and quantum diffusion by leveraging exponential growth properties of solutions. Exact results, a general proof of exponential growth, singular Green function and spectrum properties, and vanishing d.c.
Some exact results obtained by now are reviewed. A most general proof is given for the exponential growth of particular solutions. Based on this property a unification is brought about into the theories on localization of eigenstates, transmission problem, heat conduction through harmonic chains, and quantum diffusion problem. Singular properties are proved for the Green function and the spectrum of the disordered system. Vanishing of the d.c. conductivity is discussed and numerically calculated eigenstates are collected.