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

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.