Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 4 (2014), 903-927.

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

Valeria Banica and Liviu Ignat

Full-text: Open access

Abstract

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.

Article information

Source
Anal. PDE, Volume 7, Number 4 (2014), 903-927.

Dates
Received: 3 December 2012
Revised: 17 February 2014
Accepted: 1 April 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731542

Digital Object Identifier
doi:10.2140/apde.2014.7.903

Mathematical Reviews number (MathSciNet)
MR3254348

Zentralblatt MATH identifier
1297.35200

Subjects
Primary: 35B45: A priori estimates 35J10: Schrödinger operator [See also 35Pxx] 35R05: Partial differential equations with discontinuous coefficients or data 35CXX 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces)

Keywords
Schrödinger equation on metric graphs with 1-D delta potentials representation of solutions dispersion and Strichartz estimates

Citation

Banica, Valeria; Ignat, Liviu. Dispersion for the Schrödinger equation on the line with multiple Dirac delta potentials and on delta trees. Anal. PDE 7 (2014), no. 4, 903--927. doi:10.2140/apde.2014.7.903. https://projecteuclid.org/euclid.apde/1513731542


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