The pseudospectrum of the convention-diffusion operator with a variable reaction term

In this paper, we study the spectrum of non-self-adjoint convection-diffusion operator with a variable reaction term defined on an unbounded open set $\Omega$ of $\mathbb{R}^n.$ Our idea is to build a family of operators that have the same convection-diffusion-reaction formula, but which will be defined on bounded open sets $\left\{\Omega_{\eta}\right\}_{\eta\in\left]0,1\right[}$ of $\mathbb{R}^n.$ Based on the relationships that link this family to $\Omega,$ we obtain relations between the spectrum and the pseudospectrum. We use the notion of the pseudospectrum to build relationships between convection-diffusion operator and its restrictions to bounded domains. Using these relationships we are able to find the spectrum of our operator in $\mathbb{R}^+.$ Also, the techniques developed to obtain the spectrum allow us to study the properties of the spectrum of this operator when we go to the limit as the reaction term tends to zero. Indeed, we show a spectral localization result for the same convection-diffusion-reaction operator when a perturbation is carried on the reaction term and no longer on the definition domain.