Spectral analysis of the d-bar-Neumann operator

This project is located at the intersection of several different fields : complex analysis, partial differential equations, functional analysis, operator theory, spectral analysis, potential theory and mathematical physics. The main theme and the origin of the program is the d-bar-Neumann problem which ties together the analysis of several complex variables with analysis, geometry, and potential theory. Many modern techniques in Several Complex Variables have their roots in the analysis of the d-bar-Neumann problem, and the problem itself has opened up whole new fields during the development of tools for its analysis. It is planned to concentrate on the problem of compactness of the d-bar-Neumann operator using an approach which has the advantage that it covers both bounded pseudoconvex domains as well as unbounded domains with weights and which can therefore be used to handle unsolved problems for unbounded domains. Our goal will be to obtain precise conditions under which the complex Laplacians on (0,q)-forms are with compact resolvent. For this purpose we will use a formula from the d-bar-Neumann problem, which opens interesting aspects in differential geometry, and a recent necessary and sufficient condition for compactness of the corresponding d-bar-Neumann operator on (0,q)-forms. We will also analyze the spectrum of the d-bar-Neumann Laplacian and we will apply these methods for the spectral analysis of certain Schrödinger, Dirac and Pauli operators. In addition we will consider the canonical solution operator to the d-bar equation. Interestingly, in many situations, the restriction of the canonical solution operator of d-bar to forms with holomorphic coefficient arises naturally. These restrictions can be seen as certain Hankel operators on Bergman spaces of holomorphic functions. There is an interplay between the geometry of the boundary of a domain and compactness of Hankel and Toeplitz operators with symbols decaying on the boundary. We plan to analyse properties of the commutators [P, M_j], where P is the Bergman projection and M_j are the multiplication operators by the coordinate functions, which are related to the question of compactness of the d-bar-Neumann operator. Recent results on higher order forms (Celik and Sahutoglu) encourage us to attack the still open questions for (0,1)-forms. The funds of the project will be used for travel costs and to support one PhD student and one Postdoc. We plan to use funds of already approved projects of WTZ (program Amadee) in our Complex Analysis group in Vienna to supplement these activities. In November 2015 we will organize a workshop at the Erwin Schrödinger Institute (Vienna) in order to gain international platforms for further supporting discussions of the topic and to start disseminating the results of the project.

Project: P 28154-N13 Spectral analysis of the d-bar-Neumann operator The project was located at the intersection of several different fields : complex analysis, partial differential equations, functional analysis, operator theory, spectral analysis, potential theory, differential geometry, and mathematical physics. The d-bar Neumann operator - the solution operator for the complex Laplacian defined on complex differential forms - provides an important tool to describe analytic and geometric aspects of the Cauchy-Riemann equations for several complex variables. Considering weighted L2- spaces, an interesting connections to certain Schrodinger, Dirac and Pauli operators becomes apparent. In addition, the project leader observed that a similar situations appears choosing a different underlying Hilbert space, the Segal Bargmann space, and replacing the d-bar operator by the d operator. In this way, the investigation on d-complex starts with its ramifications to the creation and annihilation operator of quantum mechanics. During the time of the project the project leader finished a monograph with the title "Complex Analysis- a functional analytic approach", which appeared 2018 in the series De Gruyter Graduate. Franz Berger, supported by the project, finished his PhD studies 2018. Two postdocs joined the program, bringing many new inspiring aspects. The project leader gave many invited talks for conferences and seminars: Doha (Texas A& M University), American University Beirut, University of Ufa (Russia), Steklov Institute (Moscow), University of Sao Paulo (Brazil), Columbia University New York, Rutgers University Philadelphia, AMS-meeting Honolulu, BIRS Conference Ban (Canada), University of Brno , CIRM Conference Luminy. The international contacts where enhanced by WTZ-OAD projects Montenegro. In 2018 the workshop "Analysis and CR Geometry" at the Erwin Schrodinger Institute was organized by the complex analysis group of Vienna with about 30 leading experts from all over the world.