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Session 5. Complex Analysis
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Spectral properties of the \(\overline{\partial}\)-Neumann operator |
Friedrich Haslinger, University of Vienna, Austria
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We consider the \(\overline{\partial}\)-Neumann operator
\[
N : L^2_{(0,q)}(\Omega ) \longrightarrow L^2_{(0,q)}(\Omega ),
\]
where \(\Omega \subset \mathbb{C}^n\) is bounded pseudoconvex
domain, and
\[
N_\varphi : L^2_{(0,q)}(\Omega , e^{-\varphi}) \longrightarrow
L^2_{(0,q)}(\Omega , e^{-\varphi}),
\]
where \(\Omega \subseteq \mathbb{C}^n\) is a pseudoconvex domain and
\(\varphi \) is a plurisubharmonic weight function. \(N\) is the
inverse to the complex Laplacian \(\Box = \overline{\partial} \,
\overline{\partial}^* + \overline{\partial}^* \,
\overline{\partial}.\)
Using a general description of precompact subsets in \(L^2\)-spaces
we obtain a characterization of compactness of the
\(\overline{\partial}\)-Neumann operator, which can be applied to
related questions about Schrödinger operators with magnetic field
and Pauli and Dirac operators and to the complex Witten Laplacian.
In this connection it is important to know whether the Fock space
\[
\mathcal{A}^2 (\mathbb{C}^n, e^{-\varphi }) =\{ f\colon \mathbb{C}^n
\longrightarrow \mathbb{C} \ {\text{entire}} :
\int_{\mathbb{C}^n} |f|^2 e^{-\varphi } \, d\lambda < \infty \}
\]
is infinite-dimensional, which depends on the behavior at infinity
of the eigenvalues of the Levi matrix of the weight function
\(\varphi\).
In addition we discuss obstructions to compactness of the
\(\overline{\partial}\)-Neumann operator, and we describe, in some
special cases, the spectrum of the \(\Box\)-operator.
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References- F. Haslinger, Compactness for the
\(\overline{\partial}\)-Neumann problem - a functional analysis
approach, Collectanea Mathematica 62 (2011), 121--129.
- F. Haslinger, Spectrum of the
\(\overline{\partial}\)-Neumann Laplacian on the Fock space, J. of
Math. Anal. and Appl. 402 (2013), 739--744.
- F. Haslinger, Sobolev inequalities and the
\(\overline{\partial}\)-Neumann operator, J. of Geom. Anal., to
appear.
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