Distribution function of the local density of states of a one-channel   weakly disordered ring in an external magnetic field
  A real space diagrammatic method, which is an extension of the Berezinskii technique to problems with periodic boundary condition, is formulated to study the density of states (DOS) \rho(\epsilon,\phi) and its moments for a one-channel weakly disordered ring threaded by an external magnetic flux \phi. The exact result obtained for the average value of the DOS shows that \rho(\epsilon,\phi) oscillates with a period of the flux quantum \phi_0=hc/e. However all higher moments of the DOS oscillate with the halved period \phi_0/2. The exact expression for the DOS is valid for both weak localization (L >> l, where L is the rings circumference and l is the mean free path) and ballistic (L < l) regimes. In the weak localization regime the distribution function of the DOS is calculated, which turns out to be of logarithmic normal form. 
H. Feldmann, E. P. Nakhmedov and R. Oppermann