Eigenvalue distribution for non-self-adjoint operators on compact manifolds with small multiplicative random perturbations (Q1959937)

The author considers a semi-classical partial differential operator on a compact manifold, locally given by \[ P= \sum_{|\alpha|\leq m} a_\alpha(x; h) (hD)^\alpha, \] with \(h\in ]0,h_0]\), \(0< h_0\ll 1\). The operator is elliptic and non-self-adjoint, submitted to a small muliplicative random perturbation. Precise bounds are given for the number of the eigenvalues in subdomains of \(\mathbb{C}\), as version for compact manifolds of the results of the author [Ann. Fac. Sci. Toulouse, Math. (6) 18, No. 4, 739--795 (2009; Zbl 1194.47058)] for operators in \(\mathbb{R}^n\).