1211.7281 (V. Banica et al.)

Dispersion for the Schrödinger equation on the line with multiple
Dirac delta potentials and on delta trees    [PDF]

V. Banica, L. I. Ignat

In this paper we consider the time dependent one-dimensional Schr\"odinger equation with multiple Dirac delta potentials {of different positive strengths}. We prove that the classical dispersion property holds. 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 Kirchoff conditions, here the resolvent is no longer in the framework of Wiener algebra of almost periodic functions, and its expression is harder to analyze.

View original: http://arxiv.org/abs/1211.7281