Beyond the classical Weyl and Colin de Verdière’s formulas for Schrödinger operators with polynomial magnetic and electric fields

Boyarchenko, Mitya; Levendorski, Sergei

Annales de l'Institut Fourier, Tome 56 (2006) no. 6, pp. 1827-1901.

Nous donnons deux formules conjecturelles pour calculer le terme dominant du comportement asymptotique du spectre d’un opérateur de Schrödinger agissant dans $L^2\left({ℝ}^n\right)$ avec des polynômes quasi-homogènes comme champs électriques et magnétiques. La construction se base sur la méthode des orbites de Kirillov, et s’applique donc à n’importe quelle algèbre de Lie nilpotente. Elle est liée à la géométrie des orbites coadjointes et à certaines “intégrales algébriques” étudiées par Nilsson. En utilisant la méthode de variation directe, nous démontrons que nos formules sont correctes non seulement dans le cas régulier où s’appliquent les formules de Weyl ou Colin de Verdière, mais aussi dans certains cas “irréguliers” avec différents types de dégéréscence des potentiels.

We present a pair of conjectural formulas that compute the leading term of the spectral asymptotics of a Schrödinger operator on $L^2\left({ℝ}^n\right)$ with quasi-homogeneous polynomial magnetic and electric fields. The construction is based on the orbit method due to Kirillov. It makes sense for any nilpotent Lie algebra and is related to the geometry of coadjoint orbits, as well as to the growth properties of certain “algebraic integrals,” studied by Nilsson. By using the direct variational method, we prove that the formulas give the correct answer not only in the “regular” cases where the classical formulas of Weyl or Colin de Verdière are applicable but in many “irregular” cases, with different types of degeneration of potentials.

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