Étude spectrale d’opérateurs sur les groupes nilpotents. (Spectral study of operators on nilpotent groups).(French)Zbl 0733.35086

Sémin. Équations Dériv. Partielles 1989-1990, No. 18, 8 p. (1990).
Let $$\pi (X_ 1),...,\pi (X_ p)$$ p differential operators of order $$\leq 1$$ of the form $\pi (X):=A_ 1\frac{\partial}{\partial x_ 1}+A_ 2(x_ 1)\frac{\partial}{\partial x_ 2}+...+A_ n(x_ 1,...,x_{n-1})\frac{\partial}{\partial x_ n}+iB(x),$ where the $$A_ j$$ and B are real polynomials. The authors consider the operator $$\pi (P):=-\sum^{p}_{j=1}\pi (X_ j)^ 2.$$
Under the given assumptions $$\pi$$ (P) is selfadjoint and has compact resolvent. - Generalizing results of the second and the third author [J. Math. Pures Appl., IX. Sér. 70, No.1, 87-99 (1991; Zbl 0725.35068)] a two-sided estimation is given for the number of eigenvalues N($$\lambda$$) of $$\pi$$ (P) which are $$\leq \lambda$$. According to the specific assumptions the results can also be formulated in an algebraic way.
The proof is roughly indicated in one sentence.

MSC:

 35P20 Asymptotic distributions of eigenvalues in context of PDEs 35P15 Estimates of eigenvalues in context of PDEs 35G05 Linear higher-order PDEs

Zbl 0725.35068
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