Dispersion for the Schr{\"o}dinger equation on the line with short-range array of delta potentials

TL;DR

This paper investigates the dispersive behavior of the one-dimensional Schrödinger equation with a sequence of delta potentials, establishing decay estimates under certain conditions.

Contribution

It provides explicit dispersive estimates for the Schrödinger operator with short-range delta array potentials, extending understanding of their long-time behavior.

Findings

Established L^1 to L^∞ dispersive estimate with |t|^{-1/2} decay rate.

Derived resolvent kernel representations using Jost solutions and Born series.

Proved decay estimates assuming no zero-energy resonance and decay of coupling constants.

Abstract

We study dispersive properties of the one-dimensional Schr{\"o}dinger equation with a short-range array of delta interactions. More precisely, we consider the self-adjoint operator obtained by perturbing the free Laplacian on the line with a real-valued sequence of Dirac delta potentials and belonging to weighted ${\ell}$^1(Z) spaces. Under suitable decay assumptions on the coupling constants and in the absence of a zero-energy resonance, we establish the L^1 (R) $\rightarrow$ L^$\infty$ (R) dispersive estimate with decay rate |t|^{-1/2} for the associated Schr{\"o}dinger group. The proof relies on a limiting absorption principle in weighted spaces, explicit representation of the resolvent kernel in terms of Jost solutions and Born series expansion of the Friedrichs extension of the perturbed operator.