Asymptotics of the spectrum of pseudo-differential operator with periodic bicharacteristics

Yu. G. Safarov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 18, Tome 152 (1986), pp. 94-104

Résumé

Let $\lambda_j$ be the eigenvalues of positive elliptic pseudodifferential operator of order $m>0$ on compact closed $d$-dimentional $C^\infty$-manifold, $N(\lambda)=\sharp\{j:\lambda_j\leqslant\lambda^m\}$. It is shown that for each $\varepsilon>0$ \begin{gather*} c_0(\lambda+\varepsilon)^d+c_1\lambda^{d-1}+Q(\lambda+\varepsilon)\lambda^{d-1}+o(\lambda^{d-1})\geqslant N(\lambda)\geqslant\\ \geqslant c_0(\lambda-\varepsilon)^d+c_1\lambda^{d-1}+Q(\lambda-\varepsilon)\lambda^{d-1}+o(\lambda^{d-1}), \end{gather*} where $c_0$ and $c_1$ are standard Weyl constants, $Q(\mu)$ is some bounded function on $\mathbb R^1$. The function $Q(\mu)$ describes the influence of periodic bicharacteristics on the asymptotics of $N(\lambda)$. Under assumption of simple reflection of bicharacteristics this result is valid for differential operators on compact manifold with boundary too.