Statistical Properties of Eigenfunctions of Random Quasi 1d One-Particle Hamiltonians

The article reviews recent analytical results concerning statistical properties of eigenfunctions of random Hamiltonians with broken time reversal symmetry describing a motion of a quantum particle in a thick wire of finite length L. It is demonstrated that the problem is equivalent to the study of properties of large Random Banded Matrices in the limit of large width of the band. Matrices of this class are relevant for a number of problems in Solid State physics and in the domain of Quantum Chaos. We find the analytical expressions for the distribution of the following quantities: i) the eigenfunction amplitude |ψ(r)| 2 at given point of the sample; ii) spatial extent of the eigenfunction measured by the “inverse participation ratio” P=∫ V dr|ψ(r)| 4 ; iii) the quantity R=|ψ(r)ψ(r′)| 2 , points r and r′ belonging to the opposite ends of the sample. For a long sample the quantity –(ln R)/L characterizes the decay rate of a localized eigenfunction (Lyapunov exponent). Relation with available numerical results is discussed.