Exact solution of the time-dependent Schrödinger equation for the Bethe lattice in the presence of an impurity

The time evolution for a particle such as an electron or an exciton, moving on the Bethe lattice with one impurity is obtained exactly by solving the time-dependent Schrödinger equation via a continued fraction formalism. To examine the behavior of the time evolution influenced by the impurity, we obtain and analyze the quantum probability that a particle can be found at an initial position in the long-time limit. We show that a condition exists for the particle to be trpped by the impurity and is determined by the lattice connectivity. The condition is given exactly as a function of the impurity strength and the coordination number.