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Dispersion for the Schr{ö}dinger equation on the line with short-range array of delta potentials

Romain Duboscq, Élio Durand-Simonnet, Stefan Le Coz

Abstract

We study dispersive properties of the one-dimensional Schr{ö}dinger equation with a short-range array of delta interactions. More precisely, we consider the self-adjoint operator obtained by perturbing the free Laplacian on the line with a real-valued sequence of Dirac delta potentials and belonging to weighted ${\ell}$^1(Z) spaces. Under suitable decay assumptions on the coupling constants and in the absence of a zero-energy resonance, we establish the L^1 (R) $\rightarrow$ L^$\infty$ (R) dispersive estimate with decay rate |t|^{-1/2} for the associated Schr{ö}dinger group. The proof relies on a limiting absorption principle in weighted spaces, explicit representation of the resolvent kernel in terms of Jost solutions and Born series expansion of the Friedrichs extension of the perturbed operator.

Dispersion for the Schr{ö}dinger equation on the line with short-range array of delta potentials

Abstract

We study dispersive properties of the one-dimensional Schr{ö}dinger equation with a short-range array of delta interactions. More precisely, we consider the self-adjoint operator obtained by perturbing the free Laplacian on the line with a real-valued sequence of Dirac delta potentials and belonging to weighted ^1(Z) spaces. Under suitable decay assumptions on the coupling constants and in the absence of a zero-energy resonance, we establish the L^1 (R) L^ (R) dispersive estimate with decay rate |t|^{-1/2} for the associated Schr{ö}dinger group. The proof relies on a limiting absorption principle in weighted spaces, explicit representation of the resolvent kernel in terms of Jost solutions and Born series expansion of the Friedrichs extension of the perturbed operator.
Paper Structure (18 sections, 18 theorems, 223 equations)

This paper contains 18 sections, 18 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Functional spaces
  4. Weighted sequences
  5. Schwarz space
  6. Weighted Lebesgue and Sobolev spaces
  7. Fourier transform
  8. Hardy space
  9. Borel measures and total variation norm
  10. Bounded linear operators
  11. Schrödinger operators
  12. The free Laplace operator
  13. The Laplace operator with an infinite number of point interactions
  14. Limiting absorption principle
  15. Jost solutions
  16. ...and 3 more sections

Key Result

Theorem 1.1

Assume that there exists $\mu \in (0, 1)$ such that $(j^{1 + \mu} \alpha_j) \in \ell^1 (\mathbb Z)$ and that there is no resonance at zero energy, or that $(j^2 \alpha_j) \in \ell^1 (\mathbb Z)$. Then, for any $t \in \mathbb R^*$ and $f \in L^1 (\mathbb R)$, the dispersive estimate holds.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • proof : Proof of Theorem \ref{['thLAP']}
  • Theorem 4.1
  • ...and 27 more