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 ${\psi }_t(x,t)-\mathrm{ı}(\tau (x){\psi }_x(x,t){)}_x=0,\text{ with }x\in (0,1)$, controlled by Dirichlet-Neumann conditions. Then, exponential and polynomial decay rate of the solution are established using multiplier method.