In this work, we use semigroup theory and the energy method, together with compactness
arguments, to establish the local existence of a solution and the well-posedness
of the weighted semilinear Schrödinger equation.

In this work, we investigate the problem of exact controllability for the equation
$$u''-\Delta u=0 \quad \text{in} \quad \widehat{Q},$$
where $\hat{Q}$ is a non-cylindrical domain of $\mathbb{R}^{n+1}$. To overcome the challenges imposed by the non-cylindrical geometry, we perform an appropriate change of variables, which allows us to transform problem \eqref{Eq.Onda_} into an equivalent problem in a cylindrical domain. Thus, we study the exact controllability of the equation
$$w'' - \displaystyle\sum_{i,j=1}^n \frac{\partial}{\partial x_i} \left( a_{ij}(x,t) \frac{\partial w}{\partial x_j} \right) + \sum_{i=1}^n b_i(x,t) \frac{\partial w'}{\partial x_i} + \sum_{i=1}^n \beta_i(x,t) \frac{\partial w}{\partial x_i} = 0 \quad \text{in} \quad Q,$$
where $Q=\Omega \times (0,T)$ is a cylinder, with $\Omega \subset \mathbb{R}^n$ a bounded domain. The well-posedness results are established via the Galerkin method, ensuring existence and uniqueness of solutions. In this setting, we study the exact controllability using the Hilbert Uniqueness Method (HUM) and, finally, demonstrate the equivalence between these systems, which ensures that the controllability results obtained in the cylinder extend rigorously to the non-cylindrical domain $\hat{Q}.$