\(\overline\partial\) and Schrödinger operators
type: Item
instance of: scholarly article
title: \(\overline\partial\) and Schrödinger operators
zbMATH Open document ID: 0855.35104
DOI: 10.1007/PL00004250
author: Bo Berndtsson
published in: Mathematische Zeitschrift
publication date: 1997-02-03T00:00:00Z
full work available at URL: https://eudml.org/doc/174856
review text: We study the canonical, or \(L^2\)-minimal, solution to the \(\overline\partial\)-equation in weighted \(L^2\)-spaces in domains in \(\mathbb{C}\). It is shown that the canonical solution operator can be obtained from the Green's function for a certain Schrödinger operator. Most of the known facts about the one-dimensional \(\overline\partial\)-equation, including Hörmanders \(L^2\)-estimates, uniform estimates for Carleson-measure data, T. Wolff's theorem then follow from an inequality related to Kato's inequality, and sometimes in a sharper form. At the end we discuss some estimates for the asymptotic decay of the kernels which do not follow from Kato's inequality, and indicate the relation to the presence of a magnetic field in the Schrödinger operator.
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 922484
zbMATH Keywords: Poisson kernel
zbMATH Keywords: Bergman kernel
zbMATH Keywords: \(L^ 2\)-minimal solution
zbMATH Keywords: Schrödinger operator
MaRDI profile type: Publication
OpenAlex ID: W2327834409
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description: scientific article; zbMATH DE number 922484