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The Heisenberg oscillatorDOI: 10.1007/s13373-012-0021-z Keywords: Nilpotent Lie groups,Harmonic oscillator,Representation of nilpotent Lie groups Abstract: In this short note, we determine the spectrum of the Heisenberg oscillator which is the operator defined as $L+|x|^2+|y|^2$ on the Heisenberg group $H_1=\mathbb{ R} ^2_{x,y}\times \mathbb{ R} $ where L stands for the positive sublaplacian.
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