Monday, December 3, 2012

1211.7281 (V. Banica et al.)

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

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.
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