Dispersive decay for Schrödinger evolutions on the discrete half-line

Establish quantitative operator-norm dispersive time-decay bounds for e^{−itH}P_c associated with Schrödinger equations i∂_tψ=Hψ on ℓ^2(N), where H is self-adjoint, thereby resolving the lack of prior results on dispersive decay for Schrödinger dynamics on the discrete half-line N.

Background

In contrast to extensive results for continuous models and for discrete models on the whole line Z, the literature lacked quantitative dispersive decay estimates for Schrödinger equations posed on the discrete half-line N. The authors highlight this gap in the context of distinguishing ballistic transport (growth of position expectation) from dispersive decay (spreading and decay of maxima).

Their work addresses this gap for periodic Jacobi operators on N, but the statement underscores the broader absence of such results for general Schrödinger evolutions on the half-line, motivating the development of a comprehensive theory for other classes of self-adjoint operators on ℓ^2(N).