Dispersion for the Schrödinger equation on the line with short-range array of delta potentials
Résumé
We study dispersive properties of the one-dimensional Schrödinger 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 ℓ^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 Schrödinger 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.
Mots clés
- 35B40
- dispersive estimates
- short-range potentials
- delta interactions
- 81Q10 One-dimensional Schrödinger equation
- 47A40
- 35P05
- 1991 Mathematics Subject Classification. 35Q41
- 1991 Mathematics Subject Classification. 35Q41 35P05 35B40 47A40 81Q10 One-dimensional Schrödinger equation delta interactions short-range potentials dispersive estimates Jost solutions
- Jost solutions
- Delta potentials
- Dispersive estimates
- Schrödinger equation
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