Dispersion for the Schr\"odinger equation on the line with multiple Dirac delta potentials and on delta trees

TL;DR

This paper proves dispersion properties for the Schrödinger equation with multiple delta potentials on the line and on delta-coupled trees, extending previous results to more complex geometries and coupling conditions.

Contribution

It establishes dispersion for Schrödinger equations with delta potentials on trees, generalizing prior work and analyzing resolvent properties under new coupling conditions.

Findings

Dispersion holds under certain restrictions on potential strengths and interval lengths.

Resolved the Schrödinger equation on delta trees with complex coupling conditions.

Extended analysis of the resolvent beyond Wiener algebra framework.

Abstract

In this paper we consider the time dependent one-dimensional Schr\"odinger equation with multiple Dirac delta potentials {of different strengths}. We prove that the classical dispersion property holds under some restrictions on the strengths and on the lengths of the finite intervals. The result is obtained in a more general setting of a Laplace operator on a tree with $\delta$-coupling conditions at the vertices. The proof relies on a careful analysis of the properties of the resolvent of the associated Hamiltonian. With respect to the analysis done in \cite{MR2858075} for Kirchhoff conditions, here the resolvent is no longer in the framework of Wiener algebra of almost periodic functions, and its expression is harder to analyze.