  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.
