The Montgomery model revisited

B. Helffer¹

¹ Laboratoire de Mathématiques Université Paris-Sud and CNRS Bât. 425 F-91405 Orsay Cedex, France

Colloquium Mathematicum, Tome 118 (2010) no. 2, pp. 391-400

Résumé

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.

Détail

DOI : 10.4064/cm118-2-3

Keywords: discuss spectral properties operator mathfrak mathcal alpha frac bigg frac alpha bigg line first briefly describe operator appears various problems analysis operators nilpotent lie groups spectral properties schr dinger operator magnetic field superconductivity proof minimum alpha groundstate energy attained unique point prove minimum non degenerate study seen refinement specific nilpotent group general analysis proposed nbsp dziuba ski nbsp hulanicki nbsp jenkins