Quantum state of pendulum under exponentially increasing gravitation

We obtained the Schrödinger solution for a pendulum under exponentially increasing gravitation using dynamical invariant operator method. The quantum mechanical energy expectation value and propagator are calculated. The uncertainty relation in coherent state is exactly same with the minimum value of that in number state. We confirmed that there exists a simple relation between Berryʹs phase and Hannayʹs angle for this system.