The D-Bar Neumann Problem and Schrodinger Operators

The topic of this book is located at the intersection of complex analysis, operator theory and partial differential equations. First we investigate the canonical solution operator to d-bar restricted to Bergman spaces of holomorphic L2 functions in one and several complex variables. These operators [...]
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The topic of this book is located at the intersection of complex analysis, operator theory and partial differential equations. First we investigate the canonical solution operator to d-bar restricted to Bergman spaces of holomorphic L2 functions in one and several complex variables. These operators are Hankel operators of special type. In the following we consider the general d-bar-complex and derive properties of the complex Laplacian on L2 spaces of bounded pseudoconvex domains and on weighted L2 spaces. The main part is devoted to compactness of the d-bar-Neumann operator. The last part will contain a detailed account of the application of the d-bar-methods to Schrodinger operators, Pauli and Dirac operators and to Witten-Laplacians.