Dispersion on certain Cartesian products of graphs

TL;DR

This paper establishes a sharp dispersive estimate for the evolution operator on Cartesian products of integer lattices and finite graphs, including structures like ladders and cylinders, with potential applications.

Contribution

It proves a precise dispersive estimate for the Schrödinger evolution on specific graph products, extending understanding of quantum dynamics on complex structures.

Findings

Sharp dispersive estimate $ orm{e^{itH}f}_ < t^{-d/3} orm{f}_1$ established

Includes infinite ladder, $k$-strips, and cylinders with potentials

Results applicable to quantum dynamics on graph structures

Abstract

In this short note we prove a sharp dispersive estimate $\|\mathrm{e}^{\mathrm{i} tH} f\|_\infty < t^{-d/3}\|f\|_1$ for any Cartesian product $\mathbb{Z}^d\mathop\square G_F$ of the integer lattice and a finite graph. This includes the infinite ladder, $k$-strips and infinite cylinders, which can be endowed with certain potentials.