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There are few exactly solvable lattice models and even fewer solvable quantumlattice models. Here we address the problem of finding the spectrum of thetight-binding model (equivalently, the spectrum of the adjacency matrix) onCayley trees. Recent approaches to the problem have relied on the similaritybetween Cayley tree and the Bethe lattice. Here, we avoid to make any ansatzrelated to the Bethe lattice due to fundamental differences between the twolattices that persist even when taking the thermodynamic limit. Instead, weshow that one can use a recursive procedure that starts from the boundary andthen use the canonical basis to derive the complete spectrum of thetight-binding model on Cayley Trees. Our resulting algorithm is extremelyefficient, as witnessed with remarkable large trees having hundred of shells.We also shows that, in the thermodynamic limit, the density of states isdramatically different from that of the Bethe lattice.

Spectrum of the tight-binding model on Cayley Trees and comparison with Bethe Lattices