Étude du spectre pour un opérateur globalement elliptique dont le symbole de Weyl présente des symétries. I: Action des groupes finis. (Study of the spectrum of a global elliptic operator whose Weyl symbol shows symmetries. I: Action of finite groups). (French) Zbl 0583.58029

Helffer, Bernard; Robert, Didier

Am. J. Math. 106, 1199-1236 (1984).

The authors study the spectrum of a global elliptic pseudo-differential operator Q, whose Weyl symbol is invariant under the action of a finite group G. To each irreducible character \(\chi\) of G there corresponds a projection P of \(L^ 2({\mathbb{R}}^ n)\) such that \([Q,P]=0\). They estimate the asymptotic distribution of the eigenvalues of \(Q|_{Im P}\) in terms of \(\chi\) and the distribution of the eigenvalues of Q.

Reviewer: J.Marschall

MSC:

58J50

Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

asymptotic distribution of the spectrum; global elliptic pseudo- differential operator; Weyl symbol