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We consider equivariant solutions of Schrödinger equations on C\{0} with harmonic oscillator potentials. We determine the spaces of equivariant quantum states in three cases: for an isotropic and anisotropic harmonic oscillator potential centered at 0, and for a potential not centered at 0.

The spaces of quantum states for some harmonic oscillator potentials on the punctured plane C\{ 0 }