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

TL;DR

This paper investigates pointwise convergence and convergence rates of generalized Schrödinger operators with perturbations and polynomial growth, providing sharp results and broad applicability to various operators.

Contribution

It establishes the stability of convergence under perturbations and derives convergence rates based on phase function growth, extending results to non-elliptic operators.

Findings

Pointwise convergence remains valid under small perturbations.

Sharp convergence results for Boussinesq and Beam operators in 2D.

Convergence rate depends solely on phase function growth condition.

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.