Magnetic Schrödinger operators and the $\overline{\partial}$-equation

Abstract

In this paper we characterize compactness of the canonical solution operator to $\Bar{\partial}$ on weigthed $L^{2}$ spaces on $\mathbb{C}$. For this purpose we consider certain Schrödinger operators with magnetic fields and use a condition which is equivalent to the property that these operators have compact resolvents. We also point out what are the obstructions in the case of several complex variables.