Failure of exterior observability for a Schrödinger equation with slower surface dispersion

We consider a Schrödinger field on a planar annulus with a dynamic inner boundary and a Dirichlet outer boundary. When the surface dispersion coefficient is one half of the bulk coefficient, observation of the outer normal derivative fails to control the initial energy at every positive time. We construct smooth solutions with unit initial energy and exponentially small observation on each fixed time interval. High angular frequencies produce modes concentrated near the inner boundary. A variational construction on a short collar, followed by a comparison in the graph norm of the evolution operator, turns these localized modes into exact solutions on the full annulus.