Asymptotics of the spectrum of a pseudodifferential operator with periodic bicharacteristics. (Russian. English summary) Zbl 0621.35071
Let \(\lambda_ j\) be the eigenvalues of the positive elliptic pseudodifferential operator of order \(m>0\) on a compact closed d- dimensional \(C^{\infty}\)-manifold, \(N(\lambda)=\#\{j:\lambda_ j\leq \lambda^ m\}\). It is shown that for each \(\epsilon >0\)
\[
C_ 0(\lambda +\epsilon)^ d+C_ 1\lambda^{d-1}+Q(\lambda +\epsilon)\lambda^{d-1}+O(\lambda^{d-1})\geq N(\lambda)\geq C_ 0(\lambda -\epsilon)^ d+c_ 1\lambda^{d-1}+Q(\lambda - \epsilon)\lambda^{d-1}+O(\lambda^{d-1}),
\]
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 manifolds with boundary too.
MSC:
35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
35S05 | Pseudodifferential operators as generalizations of partial differential operators |
58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |