Klaus Gansberger
$\overline\partial$ and the Dirac Operator
Preprint series: ESI preprints

MSC:

32F20 $\overline\partial$- and $\overline\partial_b$-Neumann problems, See also {35N15}

35N15 $\overline\partial$-Neumann problem and generalizations; formal complexes, See also {32F20 and 58G05}

35J10 Schrodinger operator, See also {35Pxx}

47A10 Spectrum, resolvent

46E35 Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

Abstract: In the present paper, we prove an abstract functional analytic
criterion for an elliptic linear partial differential operator
acting on a domain in $\Bbb R^n$ to have compact resolvent.
This is applied to the $\dquer$-Neumann problem in weighted
$L^2$-spaces on $\Bbb C^n$ to obtain necessary and sufficient
conditions for existence and compactness of the $\dquer$-Neumann
operator for a class of weight functions that is more general
than the ones considered in the literature up to now.
As another application, we give some embedding Theorems for
certain weighted Sobolev spaces. Moreover, we point out the
relationship between the $\dquer$-Laplacian and the Dirac operator
in real dimension two and prove a non-compactness result for its resolvent.


Keywords: $\overline\partial$-Neumann problem, compactness, Dirac operator, Pauli operator, compact resolvent, Sobolev spaces