A study on a class of generalized Schr\"odinger operators

Abstract

In this paper, we consider the pointwise convergence for a class of generalized Schr\"{o}dinger operators with suitable perturbations, and convergence rate for a class of generalized Schr\"{o}dinger operators with polynomial growth. We show that the pointwise convergence results remain valid for a class of generalized Schr\"{o}dinger operators under small perturbations. As applications, we obtain the sharp convergence result for Boussinesq operator and Beam operator in $\mathbb{R}^2$. Moreover, the convergence result for a class of non-elliptic Schr\"{o}dinger operators with finite-type perturbations is built. Furthermore, we proved that the convergence rate for a class of generalized Schr\"{o}dinger operators with polynomial growth depends only on the growth condition of their phase functions. This result can be applied to all previously mentioned operators, and more operators.