Vol. 7, No. 4, 2014

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

Valeria Banica and Liviu I. Ignat

Vol. 7 (2014), No. 4, 903–927

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 δ-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.

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Schrödinger equation on metric graphs, with 1-D delta potentials, representation of solutions, dispersion and Strichartz estimates
Mathematical Subject Classification 2010
Primary: 35B45, 35J10, 35R05, 35CXX, 35R02
Received: 3 December 2012
Revised: 17 February 2014
Accepted: 1 April 2014
Published: 27 August 2014
Valeria Banica
Laboratoire de Mathématiques et de Modélisation d’Évry (UMR 8071)
Département de Mathématiques
Université d’Évry
23 Bd. de France
91037 Evry
Liviu I. Ignat
Institute of Mathematics “Simion Stoilow” of the Romanian Academy
21 Calea Grivitei Street
010702 Bucharest
Faculty of Mathematics and Computer Science
University of Bucharest
14 Academiei Str.
010014 Bucharest