Energy levels in a self-similar fractal cluster

The energy spectrum of atomic clusters with a fractal structure corresponding to a Sierpinski triangle on a hexagonal lattice are studied theoretically using a simple tight-binding Hamiltonian. The evolution of the energy levels and degeneracy with the growing generation of the fractal cluster is investigated. The energy states are classified into two groups: growing states and temporary states. States belonging to the first group continue to grow after appearing at a certain generation, while those of the second group do not grow. The self-similar structure of the cluster model is reflected in the growing states, which consist of three distinct types. The energy levels of the growing states, whose degeneracy obeys a recurrence relation, can be expressed by an iterated or multi-nested function including the infinitely nested square root function.