STATISTICAL PROPERTIES OF EIGENFUNCTIONS OF RANDOM QUASI 1D ONE-PARTICLE HAMILTONIANS

The article reviews recent analytical results concerning statistical p roperties of eigenfunctions of random Hamiltonians with broken time re versal symmetry describing a motion of a quantum particle in a thick w ire of finite length L. It is demonstrated that the problem is equival ent 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 Q uantum Chaos. We find the analytical expressions for the distribution of the following quantities: i) the eigenfunction amplitude \psi(r)\(2 ) at given point of the sample; ii) spatial extent of the eigenfunctio n measured by the ''inverse participation ratio'' P = integral(V) dr\p si(r)\(4); iii) the quantity R = \psi(r)psi(r')\(2), points r and r' b elonging to the opposite ends of the sample. For a long sample the qua ntity -(ln R)IL characterizes the decay rate of a localized eigenfunct ion (Lyapunov exponent). Relation with available numerical results is discussed.