Magnetic Schrödinger operators and the \(\overline{\partial}\)-equation
type: Item
instance of: scholarly article
title: Magnetic Schrödinger operators and the \(\overline{\partial}\)-equation
zbMATH Open document ID: 1123.32021
published in: Journal of Mathematics of Kyoto University
publication date: 2007-02-20T00:00:00Z
review text: The author characterizes compactness of the canonical solution of \(\widetilde\partial\) on weighted \(L^2\) spaces on \(\mathbb{C}\). Precisely if \(\varphi\) is a weight function, \(L^2_\varphi = \{f: \mathbb{C} \to \mathbb{C} \mid \|f\|^2_{\varphi^2}= \int_{\mathbb{C}} |f(x)|^2 e^{-2\varphi(z)} \,d\lambda (z) < \infty\}\) it is classical that there is a continuous solution operator \(\widetilde S : L^2_\varphi \to L^2_\varphi\) such that \(\overline\partial \widetilde S(f) = f\) [\textit{L. Hörmander}, An introduction to complex analysis in several variables. Princeton, N.J. et al.f: van Nostrand (1966; Zbl 0138.06203), \textit{M. Christ}, J. Geom. Anal. 1, No. 3, 193--230 (1991; Zbl 0737.35011)]. The solution \(S = (I - P_\varphi)\widetilde S\) is the canonical solution operator of \(\overline\partial\), such that \(S (f) \perp A^2_\varphi\) where \(A_\varphi\) is the space of entire functions belonging to \(L^2_\varphi\) and \(P_\varphi\) denotes the Bergmann projection: \(L^2_\varphi \to A^2_\varphi\). The author proves that the solution operator \(S\) to \(\overline\partial\) is compact iff there is a real valued continuous function \(\mu\) on \(\mathbb{C}\), s.t. \(\mu(z) \to \infty\) as \(|z| \to \infty\), and \(\int_{\mathbb{C}} S \varphi \overline\varphi \,d\lambda \geq \int \mu |\varphi|^2 \,d\lambda \) for all \(\varphi \in C_0^\infty (\mathbb{C})\) where \(S\) is explicitly given on \(\mathbb{C}\) in terms of the weight function. In particular, if this weight function is \(|z|^2\) then the canonical solution is not compact. On the other side, if the weight function \(\varphi\) supposed to be subharmonic and s.t. \(\Delta \varphi\) belong to the space of doubling measures (\(\mu\) is doubling if there is a constant \(C\) s.t. \(\nu (B(z, 2r)) \leq C \nu (B(z, r))\) for all \(z \in \mathbb{C}\), \(r \in \mathbb{R}^+\)) and if \(\Delta \varphi(z) \to \infty\) as \(|z| \to \infty\), then the canonical solution operator \(S \cdot L^2_\varphi \to L^2_\varphi…
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 35N15
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 5128986
zbMATH Keywords: compact canonical solution
zbMATH Keywords: Schrödinger operators
author: Fritz Haslinger
reviewed by: Gheorghe Gussi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250281775
OpenAlex ID: W1548129775
DOI: 10.1215/KJM/1250281775
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description: scientific article; zbMATH DE number 5128986