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 -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.
Keywords
Cite
@article{arxiv.1211.7281,
title = {Dispersion for the Schr\"odinger equation on the line with multiple Dirac delta potentials and on delta trees},
author = {V. Banica and L. I. Ignat},
journal= {arXiv preprint arXiv:1211.7281},
year = {2016}
}
Comments
24p, revised and extended version (repulsive strenghts and general connection conditions considered), to appear in Analysis&PDE