The watching operators method in the theory of Frenkel exciton. Novel criterion of localization and its exact calculation for the non diagonal disordered 1D chain's zero-state

A method is proposed for manipulating with diagrammatic expansion of Green's function of Frenkel's exciton random walks on the perfect lattice. The method allows one to select diagrams, to supply diagrams with factors containing information about the number of sites the diagram has passed through, etc. Simple problems related to the defect lattices are considered using the proposed method. The new criterion of localization of Frenkel exciton - the number of sites covered by the wave function - is established. The number of sites covered by the zero state of 1D non-diagonally disordered chain is studied. It is shown that this problem can be solved by calculating the random walks Green's function with modified diagrammatic expansion. By means of the developed method an exact analitical expression for the number of sites covered by the zero-state is obtained and zero-state is shown to be localized.