# Wydawnictwa / Czasopisma IMPAN / Colloquium Mathematicum / Wszystkie zeszyty

## The Montgomery model revisited

### Tom 118 / 2010

Colloquium Mathematicum 118 (2010), 391-400 MSC: Primary 35P15. DOI: 10.4064/cm118-2-3

#### Streszczenie

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.

#### Autorzy

• B. HelfferLaboratoire de Mathématiques
Université Paris-Sud and CNRS
Bât. 425
F-91405 Orsay Cedex, France
e-mail

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