Asymptotic behavior of the spectrum of pseudodifferential operators with small parameters

D. G. Vasil'ev

Sbornik. Mathematics, Tome 49 (1984) no. 1, pp. 61-72

Résumé

The eigenvalue problem $$ L(\varepsilon,h)f\equiv\varepsilon^{m_0}A_0f+\sum^l_{j=1}h_j\varepsilon^{m_j}A_jf=\lambda f. $$ is considered on an $n$-dimensional compact manifold without boundary. Here the $A_k$, $k=0,1,\dots,l$, are symmetric scalar classical pseudodifferential operators of orders $m_k$ with leading symbols $a_k(x,\xi)$, $m_0>0$, $m_0\geqslant m_k\geqslant0$, $a_0(x,\xi)>0$ and $\varepsilon$, $h_j$, $j=1,2,\dots,l$, are small real parameters with $\varepsilon>0$ and $h_j=O(\varepsilon^{1/p})$, where $p$ is a positive integer. The distribution functions $n(\lambda,L(\varepsilon,h))$ of the eigenvalues of the operator $L(\varepsilon,h)$ are studied. Let $[\Lambda_1,\Lambda_2]$ be a fixed interval of the positive half-line ($\Lambda_1>0$). An asymptotic formula with optimal relative error $O(\varepsilon)$ is obtained for $n(\lambda,L(\varepsilon,h))$ as $\varepsilon\to0$ when $\lambda\in[\Lambda_1,\Lambda_2]$. Bibliography: 10 titles.