Exact evaluation of the causal spectrum and localization properties of electronic states on a scale-free network

A nearest-neighbor tight-binding model on a tree structure is investigated. The full energy spectrum of the normalized Hamiltonian can be expressed in terms of successively increasing number of contributions at any finite step of construction of the tree, resulting in a causal chain. The degree of quantum localization of any eigenstate, measured by the inverse participation ratio (IPR), is also analytically expressed by means of terms in corresponding eigenvalue chain. The resulting IPR scaling behavior is expressed by the tails of eigenvalue chains as well.