On the eigenvalues of a class of hypo-elliptic operators. IV

Sjöstrand, Johannes

Annales de l'Institut Fourier, Tome 30 (1980) no. 2, pp. 109-169.

Soit $P$ un opérateur pseudo-différentiel classique, d’ordre $>1$, de symbole principal non-négatif, sur une variété compacte. On suppose que $P$ est hypoelliptique avec perte d’une dérivée et semi-borné inférieurement. On construit alors exp$(-tP)$, $t\ge 0$ comme un opérateur intégral de Fourier non classique et on calcule la contribution principale à la distribution asymptotique des valeurs propres de $P$. Ce travail complète une série de travaux en collaboration avec A. Menikoff.

Let $P$ be a selfadjoint classical pseudo-differential operator of order $>1$ with non-negative principal symbol on a compact manifold. We assume that $P$ is hypoelliptic with loss of one derivative and semibounded from below. Then exp$(-tP)$, $t\ge 0$, is constructed as a non-classical Fourier integral operator and the main contribution to the asymptotic distribution of eigenvalues of $P$ is computed. This paper is a continuation of a series of joint works with A. Menikoff.

[1] L. Hörmander, A class of hypoelliptic pseudodifferential operators with double characteristics, Math. Ann., 217 (1975), 165-188. | MR | Zbl

[2] J. Karamata, Neuer Beweis und Verallgemeinerung der Tauberschen Sätze etc., J. Reine u. Angew. Math., 164 (1931), 27-39. | JFM | Zbl

[3] A. Melin, Lower bounds for pseudo-differential operators, Ark. f. Math., 9 (1971), 117-140. | MR | Zbl

[4] A. Melin and J. Sjöstrand, Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem, Comm. P.D.E., 1 (1976), 313-400. | MR | Zbl

[5] A. Melin and J. Sjöstrand, A calculus for Fourier integral operators in domains with boundary and applications to the oblique derivative problem, Comm. P.D.E., 2 (1977), 857-935. | MR | Zbl

[6] A. Menikoff and J. Sjöstrand, On the eigenvalues of a class of hypoelliptic operators, Math. Ann., 235 (1978), 55-85. | MR | Zbl

[7] A. Menikoff and J. Sjöstrand, On the eigenvalues of a class of hypoelliptic operators II, Springer L. N., n°755, 201-247. | MR | Zbl

[8] A. Menikoff and J. Sjöstrand, The eigenvalues of hypoelliptic operators, III, the non semibounded case, Journal d'Analyse Math., 35 (1979), 123-150. | Zbl

[9] J. Sjöstrand, Eigenvalues for hypoelliptic operators and related methods, Proc. Inter. Congress of Math., Helsinki, 1978, 445-447.