On the stability of degenerate Schrödinger equation under boundary fractional damping

In this paper we study the well-posedness and stability of degenerate Schrödinger equation with a fractional boundary damping. First, we establish the well-posedness of the degenerate problem $ψ_t(x,t)-\imath(τ(x) ψ_x(x,t))_x=0, \hbox{ with } x \in (0,1)$, controlled by Dirichlet-Neumann conditions. Then, exponential and polynomial decay rate of the solution are established using multiplier method.