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

Abstract

In this paper we consider the time dependent one-dimensional Schrödinger 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.

24p, revised and extended version (repulsive strenghts and general connection conditions considered), to appear in Analysis&PDE

References in corpus (5)

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Cited by in corpus (5)

• Ground state and orbital stability for the NLS equation on a general starlike graph with potentials
• Bifurcations of standing localized waves on periodic graphs
• Standing waves on quantum graphs
• NLS ground states on metric trees: existence results and open questions
• Orbital stability of standing waves for supercritical NLS with potential on graphs