Dispersive estimates for matrix Schr\"{o}dinger operators in dimension two

TL;DR

This paper establishes dispersive decay estimates for a class of non-selfadjoint matrix Schrödinger operators in two dimensions, relevant for analyzing linearized focusing NLS equations around standing waves.

Contribution

It provides the first dispersive decay estimates for these operators under natural spectral assumptions, including weighted estimates with logarithmic decay.

Findings

Proved $L^1$ to $L^$ decay estimates for the evolution operator.

Derived weighted estimates with decay rate $1/|t| ext{log}^2|t|$.

Applicable to linearization of focusing NLS around standing waves.

Abstract

We consider the non-selfadjoint operator [\cH = [{array}{cc} -\Delta + \mu-V_1 & -V_2 V_2 & \Delta - \mu + 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 $L^1(\R^2)\times L^1(\R^2)\to L^\infty(\R^2)\times L^\infty(\R^2)$ 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 $w(x)=\log^2(2+|x|)$.