Steep expulsive potentials in 1D and 2D Schrödinger equations support continuous spectra of normalizable bound states, including even/odd states in 1D and vortex states in 2D.
and Promislow K., Spectral and Dynamical Stability of Nonlinear Waves , Springer-Verlag (New York, 2013)
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The continuous spectrum of bound states in expulsive potentials
Steep expulsive potentials in 1D and 2D Schrödinger equations support continuous spectra of normalizable bound states, including even/odd states in 1D and vortex states in 2D.