Chain models and their effective Hamiltonians

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Abstract
We develop a new formalism suitable for diagonalization of Hamiltonians with 2t+1 nonzero diagonals (chain models). In a systematic perturbation-like way, we get the expansions of Green and/or wave functions. They are generated by solution of a sequence of ‘‘fixed-point‘‘ quadratic equations for (t×t)-dimensional matrices. For t=1 and t=2, this solution is feasible by elementary means, so that the respective tridiagonal and pentadiagonal chain models may be considered exactly solvable in this context. In an alternative first-order variational-perturbation formulation, the method may provide the simultaneous upper and lower energy bounds for any t.

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