Dispersion for the Schr\"odinger Equation on Networks

TL;DR

This paper analyzes the Schr"odinger equation on complex network structures, providing explicit solutions and dispersive estimates that facilitate understanding of both linear and nonlinear dynamics on such networks.

Contribution

It offers an explicit solution framework for the Schr"odinger equation on tree-like networks with infinite edges and extends dispersive estimates to cases with piecewise constant coefficients.

Findings

Explicit solution description for the linear Schr"odinger equation on networks.

Dispersive estimates established for the network case.

Extension of results to networks with piecewise constant coefficients.

Abstract

In this paper we consider the Schr\"odinger equation on a network formed by a tree with the last generation of edges formed by infinite strips. We give an explicit description of the solution of the linear Schr\"odinger equation with constant coefficients. This allows us to prove dispersive estimates, which in turn are useful for solving the nonlinear Schr\"odinger equation. The proof extends also to the laminar case of positive step-function coefficients having a finite number of discontinuities.