A complete classification of threshold properties for one-dimensional discrete Schrödinger operators
Abstract
We consider the one-dimensional discrete Schrödinger operator on ℤ, and study the relation between the generalized eigenstates and the asymptotic expansion of the resolvent for the threshold 0. We decompose the generalized zero eigenspace into subspaces, some of which correspond to the bound states or the resonance states, only by their growth properties at infinity, and precisely describe the first few leading coefficients in the expansion using these subspaces. The generalized zero eigenspace we consider is the largest possible one, consisting of all solutions to the eigenequation. For the resolvent expansion, we implement the expansion scheme of Jensen–Nenciu [Rev. Math. Phys.13 (2001) 717–754] and [Rev. Math. Phys.16 (2004) 675–677] in its full generality.
