<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(v_t(x,t)+\imath (x^\alpha v_x(x,t))_x=0, \hbox { with } x \in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>ı</mi> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mi>α</mi> </msup> <msub> <mi>v</mi> <mi>x</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.333333em" /> <mtext>with</mtext> <mspace width="0.333333em" /> <mi>x</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, 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.</p>