Magnetic Bloch analysis and Bochner Laplacians
type: Item
instance of: scholarly article
title: Magnetic Bloch analysis and Bochner Laplacians
zbMATH Open document ID: 0804.58022
DOI: 10.1016/0393-0440(94)90035-3
author: H. Over
author: Joachim Asch
author: Ruedi Seiler
published in: Journal of Geometry and Physics
publication date: 1995-01-12T00:00:00Z
review text: The authors consider a particle in a magnetic field where the configuration space \(M\) is assumed to be a 2-torus and the magnetic field is assumed to satisfy an integrality condition. Then quantization can be established by choosing a Hermitian line bundle over \(M\) with connection \(\nabla\) such that the curvature of \(\nabla\) equals the magnetic field. The Hilbert space of quantum mechanical states is given as the space of \(L^ 2\) sections in this bundle and the Hamiltonian is given as the Bochner Laplacian \(\nabla^* \nabla\). As the 2-torus is not simply connected, the construction is non-unique. The authors show that the family of Bochner Laplacians arising in this construction is related to the so-called Bloch decomposition of the corresponding Hamiltonian on the covering space \(\mathbb{R}^ 2\) of \(M\). Applications are not worked out, but it is mentioned that motivation for this work comes from the quantum Hall effect.
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 81S10
zbMATH DE Number: 569290
zbMATH Keywords: magnetic field
zbMATH Keywords: 2-torus
zbMATH Keywords: Bochner Laplacian
zbMATH Keywords: Bloch decomposition
zbMATH Keywords: Hamiltonian
reviewed by: Volker Perlick
MaRDI profile type: Publication
full work available at URL: 0393-0440(94)90035-3
OpenAlex ID: W1985282912
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description: scientific article; zbMATH DE number 569290