General formalism for the spectral decomposition of the Hamiltonian in a quantum fractal network

We present a general formulation for the spectral decomposition of the Hami ltonian operator of a quantum fractal network (QFN). The QFN can be constru cted by placing artificial neurons on each site of the fractal lattice. An artificial neuron may consist of a cell of a quantum cellular automata or a quantum dot which confines a single electron. The Coulomb interaction or t he spin-spin interaction between neurons can be used to transmit signals an d perform logic operations. The local external field may be as input signal to influence output of the system. We obtain explicitly the recursive form ulae of the eigenvalues and eigenvectors between sublattices, the intertwin ing relation between the collision operator and the Hamiltonian operator by combining subdynamics and a reduced lattice approach. Furthermore, the per turbation method to obtain the spectral decomposition for the time-dependen t Hamiltonian is also discussed. Finally, as an example, we calculate the e igenvalues and eigenvectors of the Hamiltonian operator for a Sierpinski ga sket based on our formulation. Analysis of the recursive formula for the sp ectrum of the Sierpinski gasket, reveals how its spectral structure changes in boundary conditions.