In this note we establish the large time limit non-negativity of the heat kernel for a class of elliptic differential operators on closed, Riemannian manifolds, and apply this result to a problem from conformal differential geometry.

This paper presents adaptive algorithms for eigenvalue problems associated with non-selfadjoint partial differential operators. The basis for the developed algorithms is a homotopy method which departs from a wellunderstood selfadjoint problem. Apart from the adaptive grid refinement, the progress of the homotopy as well as the solution of the iterative method are adapted to balance the contrib...

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...

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 paper is an overview of aspects of the singularities of the zeta function, equivalently, of the small time asymptotics of the trace of the heat semigroup, of elliptic cone operators. It begins with a brief description of classical results for regular differential operators on smooth manifolds, and includes a concise introduction to the theory of cone differential operators. The later secti...

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

A general construction of transmutation operators is developed for selfadjoint operators in Gelfand triples. Theorems regarding analyticity of generalized eigenfunctions and Paley-Wiener properties are proved.