The complete set of Eigenstates and Eigenvalues of the nearest neighbour tight binding model on a Cayley tree with branching number b = 2 and M branching generations with open boundary conditions is derived. We find that of the N = 1 + 3(2 M − 1) total states only 3 M + 1 states are extended throughout the Cayley tree. The remaining N − (3 M + 1) states are found to be strongly localised states with finite amplitudes on only a subset of sites. In particular, there are, for M > 1, 3 × 2 M −2 surface states which are each antisymmetric combinations of only two sites on the surface of the Cayley tree and have energy eactly at E = 0, the middle of the band. The ground state and the first two excited states of the Cayley tree are found to be extended states with amplitudes on all sites of the Cayley tree, for all M . We use the results on the complete set of Eigenstates and Eigenvalues ...

中文翻译：

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Cayley树上紧密绑定模型的完整解决方案：强局部状态与扩展状态

推导出分支数为b = 2的Cayley树上最近邻紧密绑定模型的特征值和特征值的完整集合，并在具有开放边界条件的情况下生成M个分支代。我们发现在N = 1 + 3（2 M − 1）个总状态中，只有3 M + 1个状态扩展到整个Cayley树。发现其余的N-（3 M +1）个状态是仅在部分站点上具有有限幅度的强局部状态。尤其是，对于M> 1，存在3×2 M -2表面状态，每个表面状态都是Cayley树表面上仅两个位点的反对称组合，并且在带中部的E = 0处实际上具有能量。发现对于所有M，Cayley树的基态和前两个激发态都是在Cayley树的所有位点上具有振幅的扩展状态。