Quantum states of hierarchical systems

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Abstract
The quantum states of an electron in a hierarchical potential are investigated in the tightbinding approximation. The hierarchy is taken to be in the transition matrix elements, in natural analogy to the classical problem of diffusion in ultrametric structures. The energy spectrum is found to be a Cantor set, and analytical results are presented for its scaling properties. The envelope of the wave function is found to decay algebraically for certain energies and to be constant for others. The results are in excellent agreement with high-precision numerical work.