Stability of Degenerate Schrödinger Equation with Harmonic Method

This paper is devoted to study the well-posedness and stability of degenerate Schrödinger equation with a boundary control acting at the degeneracy. First, we establish the well-posedness of the degenerate problem $v_t(x,t)+\mathrm{ı}(x^{\alpha }{v}_x(x,t){)}_x=0,\text{ with }x\in (0,1)$, controlled by Dirichlet-Neumann conditions. Then, exponential and polynomial decreasing of the solution are established. This result is optimal and it is obtained using complex analysis method.