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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

Citations:

Zbl 0725.35068
Full Text: Numdam EuDML