Pauli operators and the $\overline\partial$-Neumann problem

Аннотация

We apply methods from complex analysis, in particular the $\overline{\partial}$-Neumann operator, to study spectral properties of Pauli operators. For this purpose we consider the weighted $\overline{\partial}$-complex on $\mathbb{C}^n$ with a plurisubharmonic weight function. The Pauli operators appear at the beginning and at the end of the weighted $\overline{\partial}$-complex. We use the spectral properties of the corresponding $\overline{\partial}$-Neumann operator to answer the question when the Pauli operators are with compact resolvent. It is also of importance to know whether the related Bergman space of entire functions is of infinite dimension. The main results are formulated in terms of the properties of the Levi matrix of the weight function. If the weight function is decoupled, one gets additional informations. Finally, we point out that a corresponding Dirac operator fails to be with compact resolvent.