Under smoothness and growth assumptions on f we show that a standing wave w(t,x)=eibtf(x) of the Schrödinger equation on the exterior W of a ball and Neumann boundary condition wt=i(Dw +f(|w|2)w) ¶w/¶n=0 on ¶W where b is real and f is real and radially symmetric, is always linearly unstable under perturbations in the space H1(W) (it may be stable under perturbations in H1rad(W)).
The instability is independent of f having a fixed sign andof its Morse index.
The main tool is a theorem of linearized instability of M. Grillakis.
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