## The $\bar\partial$-Neumann Problem

## Basic Information

Funding Source: Austrian Science Fund (FWF)

Duration: 01.09.2011 - 31.08.2016

Principal Investigator: Friedrich Haslinger

Project Number: P23664 Stand-Alone-Project

## Description

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 $\bar \partial$-Neumann operator for bounded pseudoconvex domains. It turned out that this abstract approach was very helpful in some different situations, for instance to describe compactness of the $\bar\partial$-Neumann operator 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 $\bar\partial$. 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.

# Publications

*"On some spectral properties of the weighted $\bar\partial$-Neumann problem"*, ArXiv e-prints., September, 2015. Haslinger F (2016),

*"Sobolev Inequalities and the $\bar\partial$-Neumann Operator"*, J. Geom. Anal.. Vol. 26(1), pp. 287-293. [DOI] [URL] Haslinger F (2014),

*"The $\bar\partial$-Neumann problem and Schrödinger operators"*Vol. 59, pp. xii+241. De Gruyter, Berlin. [DOI] [URL] Haslinger F (2013),

*"Spectrum of the $\bar\partial$-Neumann Laplacian on the Fock space"*, J. Math. Anal. Appl.. Vol. 402(2), pp. 739-744. [DOI] [URL]