Narrow Results

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.

We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper, we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.

We describe a classification of threshold properties for one-dimensional discrete Schrödinger operators and associated asymptotic expansions of the resolvent for a general class of interactions.