The asymptotic distribution of eigenvalues of self-adjoint differential operators in the
high-energy limit or the semi-classical limit is a classical subject going back to H. Weyl of
more than a century ago. In the last decades there has been a renewed interest in
non-self-adjoint differential operators which have many subtle properties such as instability
under small perturbations. Quite remarkably when adding small random perturbations to such
operators the eigenvalues tend to distribute according to Weyl's law (quite differently from
the distribution for the unperturbed operators in analytic cases). A first result in this
direction was obtained by M. Hager in her thesis of 2005. Since then further general results
have been obtained which are the main subject of the present book. Additional themes from the
theory of non-self-adjoint operators are also treated. The methods are very much based on
microlocal analysis and especially on pseudodifferential operators. The reader will find a
broad field with plenty of open problems.
