On some spectral properties of the weighted ∂¯-Neumann operator

Abstract

We study necessary conditions for compactness of the weighted ∂-Neumann operator on the space L 2 (C n , e −ϕ ) for a plurisubharmonic function ϕ. Under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension, a weaker result is obtained by simpler methods. Moreover, we investigate (non-) compactness of the ∂-Neumann operator for decoupled weights, which are of the form ϕ(z) = ϕ1(z1) + · · · + ϕn(zn). More can be said if every ∆ϕj defines a nontrivial doubling measure.2010 Mathematics Subject Classification. Primary 32W05; Secondary 30H20, 35N15, 35P10.

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

“…We will also consider special weight functions, the so-called decoupled weights, and, using the tensor product structure of the essential spectrum σ e ( (0,q) ϕ ) we get the following (see [1])…”