# Observability of a 1D Schr\"odinger equation with time-varying boundaries

## TL;DR

This paper investigates the observability and controllability of a 1D Schrödinger equation with moving boundaries, providing exact conditions for observability in both linear and general moving cases.

## Contribution

It offers new precise observability results for Schrödinger equations with time-dependent domains, including boundary and internal observability conditions.

## Key findings

- Exact boundary and pointwise internal observability for linear moving boundaries.
- Boundary observability conditions for general moving boundaries under certain curve conditions.
- Controllability of the adjoint system established via duality theory.

## Abstract

We discuss the observability of a one-dimensional Schr\"odinger equation on certain time dependent domain. In linear moving case, we give the exact boundary and pointwise internal observability for arbitrary time. For the general moving, we provide exact boundary observability when the curve satisfies some certain conditions . By duality theory, we establish the controllability of adjoint system.