  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.
