Dispersive estimates for matrix Schrödinger operators in dimension two

Abstract

We consider the non-selfadjoint operator [\cH = [{array}{cc} -Δ+ μ-V_1 & -V_2 V_2 & Δ- μ+ V_1 {array}]] where $\mu >0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain $L^1({\mathbb{R}}^2)\times L^1({\mathbb{R}}^2)\to L^{\infty }({\mathbb{R}}^2)\times L^{\infty }({\mathbb{R}}^2)$ dispersive decay estimates for the evolution $e^{it\cH}P_{ac}$. We also obtain the following weighted estimate $$ \|w^{-1} e^{it\cH}P_{ac}f\|_{L^\infty(\R^2)\times L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)\times L^1(\R^2)},\,\,\,\,\,\,\,\, |t| >2, $$ with $w(x)={\log }^2(2+|x|)$.

arXiv admin note: text overlap with arXiv:1202.0050

References in corpus (3)

• Dispersive estimates for Schrodinger operators in dimensions one and three
• A counterexample to dispersive estimates for Schrödinger operators in higher dimensions
• A weighted dispersive estimate for Schrödinger operators in dimension two

Cited by in corpus (3)

• The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions
• Dispersive estimates for massive Dirac operators in dimension two
• Dispersive Estimates in R^3 with Threshold Resonances