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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 |
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Vol. 7 (2014), No. 4, 903–927
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 35B45, 35J10, 35R05, 35CXX, 35R02
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Milestones
Received: 3 December 2012
Revised: 17 February 2014
Accepted: 1 April 2014
Published: 27 August 2014
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Authors |
| Valeria Banica | |
| Laboratoire de Mathématiques et de
Modélisation d’Évry (UMR 8071) Département de Mathématiques Université d’Évry 23 Bd. de France 91037 Evry France |
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| Liviu I. Ignat | |
| Institute of Mathematics “Simion
Stoilow” of the Romanian Academy 21 Calea Grivitei Street 010702 Bucharest Romania |
Faculty of Mathematics and Computer
Science University of Bucharest 14 Academiei Str. 010014 Bucharest Romania |
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