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Abstract

We discuss the spectral properties of the operator $$ {\mathfrak h} _{\mathcal M}(\alpha):=-\frac{d^2}{dt^2} + \bigg(\frac{1}{2}\, t^{2} -\alpha\bigg)^2 $$ on the line. We first briefly describe how this operator appears in various problems in the analysis of operators on nilpotent Lie groups, in the spectral properties of a Schrödinger operator with magnetic field and in superconductivity. We then give a new proof that the minimum over $\alpha$ of the groundstate energy is attained at a unique point and also prove that the minimum is non-degenerate. Our study can also be seen as a refinement for a specific nilpotent group of a general analysis proposed by J. Dziubański, A. Hulanicki and J. Jenkins.