Dispersive estimates for matrix schrödinger operators in dimension two

M. Burak Erdoǧan; William R. Green

doi:10.3934/dcds.2013.33.4473

Dispersive estimates for matrix schrödinger operators in dimension two

M. Burak Erdoǧan, William R. Green

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the non-selfadjoint operator H = [ -δ + μ - V ₁-V₂ V₂ δ - μ + V₁ ] where μ < 0 and V₁, V₂ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain L ¹(R²)×L¹(R²) →; L ^∞(R²)→;L^∞(R²) dispersive decay estimates for the evolution e^<HP_ac. We also obtain the following weighted estimate ||w^-1e ^ithP^acf||L^∞(R²)<L ¹(R²) . 1 |t| log²(|t|) |wf|L ¹(R²)×L¹(R²), |t| > 2; with w(x) = log²(2 + |x|).

Original language	English (US)
Pages (from-to)	4473-4495
Number of pages	23
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	33
Issue number	10
DOIs	https://doi.org/10.3934/dcds.2013.33.4473
State	Published - Oct 2013

Keywords

Asymptotic stability
Dispersive estimates
Matrix Schrödinger operators
Solitons
Weighted estimates

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcds.2013.33.4473

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title = "Dispersive estimates for matrix schr{\"o}dinger operators in dimension two",

abstract = "We consider the non-selfadjoint operator H = [ -δ + μ - V 1-V2 V2 δ - μ + V1 ] where μ < 0 and V1, V2 are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain L 1(R2)×L1(R2) →; L ∞(R2)→;L∞(R2) dispersive decay estimates for the evolution e<HPac. We also obtain the following weighted estimate ||w-1e ithPacf||L∞(R2)1(R2) . 1 |t| log2(|t|) |wf|L 1(R2)×L1(R2), |t| > 2; with w(x) = log2(2 + |x|).",

keywords = "Asymptotic stability, Dispersive estimates, Matrix Schr{\"o}dinger operators, Solitons, Weighted estimates",

author = "Erdoǧan, {M. Burak} and Green, {William R.}",

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T1 - Dispersive estimates for matrix schrödinger operators in dimension two

AU - Erdoǧan, M. Burak

AU - Green, William R.

PY - 2013/10

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N2 - We consider the non-selfadjoint operator H = [ -δ + μ - V 1-V2 V2 δ - μ + V1 ] where μ < 0 and V1, V2 are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain L 1(R2)×L1(R2) →; L ∞(R2)→;L∞(R2) dispersive decay estimates for the evolution e<HPac. We also obtain the following weighted estimate ||w-1e ithPacf||L∞(R2)1(R2) . 1 |t| log2(|t|) |wf|L 1(R2)×L1(R2), |t| > 2; with w(x) = log2(2 + |x|).

AB - We consider the non-selfadjoint operator H = [ -δ + μ - V 1-V2 V2 δ - μ + V1 ] where μ < 0 and V1, V2 are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain L 1(R2)×L1(R2) →; L ∞(R2)→;L∞(R2) dispersive decay estimates for the evolution e<HPac. We also obtain the following weighted estimate ||w-1e ithPacf||L∞(R2)1(R2) . 1 |t| log2(|t|) |wf|L 1(R2)×L1(R2), |t| > 2; with w(x) = log2(2 + |x|).

KW - Asymptotic stability

KW - Dispersive estimates

KW - Matrix Schrödinger operators

KW - Solitons

KW - Weighted estimates

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Dispersive estimates for matrix schrödinger operators in dimension two

Abstract

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