We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational...

This paper touches upon several traditional topics of 1D linear complex analysis including distribution of zeros of entire functions, completeness problem for complex exponentials and for other families of special functions, some problems of spectral theory of selfadjoint differential operators. Their common feature is the close relation to the theory of complex Fourier transform of compactly s...

Some recent inequalities for the Čebyšev functional of two functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved functions and operators, are surveyed.

On utilizing the spectral representation of selfadjoint operators in Hilbert spaces, some approximations for the n-time di¤erentiable functions of selfadjoint operators in Hilbert spaces by two point Taylor’s type expansions are given. 1. Introduction Let U be a selfadjoint operator on the complex Hilbert space (H; h:; :i) with the spectrum Sp (U) included in the interval [m;M ] for some real n...

We define a new topology, weaker than the gap topology, on the space of selfadjoint operators affiliated to a semifinite von Neumann algebra and define the real-valued spectral flow for a continuous path of selfadjoint Breuer-Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips’ analytical definition. Furthermore we prove the homotopy in...

This is the first in a series of works devoted to small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength ǫ of the perturbation is ≫ h (or sometimes only ≫ h2) and bounded from above by hδ for some δ > 0. We get a complete asymptotic descri...

This is an unedited transscript of the handwritten notes of my talk at the Master Class.