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
-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 our earlier analysis for
Kirchhoff conditions [J. Math. Phys. 52:8 (2011), #083703], here the resolvent is no
longer in the framework of Wiener algebra of almost periodic functions, and its
expression is harder to analyse.
Keywords
Schrödinger equation on metric graphs, with 1-D delta
potentials, representation of solutions, dispersion and
Strichartz estimates