The d-bar Neumann Problem

This project is located at the intersection of complex analysis in several variables and spectral theory of Schrödinger operators and Hankel operators, including potential theoretic aspects. Using a description of precompact subsets in L^2-spaces we recently gave an abstract characterization of compactness of the d-bar Neumann operator for bounded pseudoconvex domains. It turned out that this abstract approach was very helpful in some different sitautions, for instance to describe compactness of the d-bar Neumann opertaor on singular spaces. It will be used as a "red thread" for this project. In this connection the potential-theoretic condition (P) is of great importance, it assumes the existence of a uniformly bounded family of functions, for which the complex Hessians are large at the boundary of the domain in consideration. Property (P) has interesting consequences for the theory of operators, which are linked in a natural way with the domain. There are also connections with the Monge- Ampere equations. These topics will be content of cooperations with colleagues from the universities of Cracow and Nice. It is of special interest to clarify if or to what extent compactness of the restriction of forms with holomorphic coefficients already implies compactness of the original solution operator to d-bar. In this connection the commutators of the Bergman projection with the coordinate functions play an important role. Spectral properties of the Witten Laplacian were used in order to find a sufficient condition for compactness of the canonical solution operator to d-bar on a weighted L^2-space. We also want to investigate the spectrum and the essential spectrum of the box operator. This is the starting point of a variety of interesting interplays between spectral theory of Pauli and Schrödinger operators and complex analysis. The recent, more functional analytic approach mentioned above will be helpful and important in order to characterize compactness of the d-bar Neumann operator on a weighted L^2- space. Another natural question is: does compactness imply exact regularity in the context of the weighted Sobolev spaces? Klaus Gansberger began to follow these ideas in his thesis and proved many promising results at the intersection of complex analysis and spectral theory of Schrödinger and Dirac operators. In this connection a continuation of the promising collaboration with Bernard Helffer will be an important point for this proposal. The spectral-theoretic questions here are often related to the problem whether certain weighted Hilbert spaces of entire functions are of infinite dimension, which would imply that 0 belongs to the essential spectrum of a corresponding Pauli operator. From this interplay between several different fields we expect to obtain new tools and new directions for the single topics.

The project was 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 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 connection to certain Schrödinger, Dirac and Pauli operators becomes apparent.The first result of the project was the complete determination of the spectrum of the complex Laplacian on the Fock space. In addition it was shown that certain properties of the boundary of a domain imply that the corresponding d-bar Neumann operator can be extended as a continuous operator on larger Lp -spaces. During the time of the project the project leader finished a monograph with the title The d-bar Neumann problem and Schrodinger operators, which appeared 2014 in the series De Gruyter Expositions in Mathematics.Four master theses were written under the supervision of the project leader, all candidates were supported by the funds of the project. Franz Berger continued as a PhD student and in the last year of the project two post docs joined the program.The project leader gave many invited talks for conferences and seminars in France, England, Poland, Italy, Spain, Norway, Turkey, China, Mexico and USA. The international contacts were enhanced by WTZ-Ö AD projects with France (Nice, Marseille), Poland (Krakow) and Slovenia (Ljubljana).In 2015 the workshop Several Complex Variables and CR Geometry at the Erwin Schrodinger Institute was organized by the members of the project with about 30 leading experts from all over the world.Some members of the project, together with the leader, continue the work in a new approved FWF grant, which started in April 2016.1