Eigenvalue counting estimates for a class of linear spectral pencils with applications to zero modes
type: Item
instance of: scholarly article
title: Eigenvalue counting estimates for a class of linear spectral pencils with applications to zero modes
zbMATH Open document ID: 1251.47016
author: Daniel M. Elton
author: Tạ Ngọc Trí
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-05-04T00:00:00Z
review text: Let \(P, Q\) be functions from \({\mathbb R}^d\) to the space of \(m\times m\) matrices. The main assumption is that \(Q\) is an \(L^p\) function (using the matrix operator norm for its range) for some \(1<p<\infty\), while the pointwise inverse of \(P\) is a weak-\(L^p\) function. The authors consider the pencil of operators \(P(-i \nabla)-zQ(x)\), \(z\in {\mathbb C}\), on \(L^2({\mathbb R}^d,{\mathbb C}^m)\), where \(Q(x)\) is a multiplication operator and \(P(-i \nabla)\) is a multiplication operator on the Fourier space. It is shown that the total number of eigenvalues with \(|z|\leqslant R\) is bounded by \(C[||P^{-1}||^\ast_{L^P_w}||Q||_{L^p}R]^p\). A motivating example for this study is the Weyl-Dirac operator, a corollary of the main theorem estimates the frequency with which zero modes occur when the magnetic potential is scaled.
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 6030456
zbMATH Keywords: linear spectral pencil
zbMATH Keywords: eigenvalue counting function
zbMATH Keywords: Dirac operator
zbMATH Keywords: zero modes
reviewed by: Vladimir V. Kisil
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.03.001
OpenAlex ID: W2022373004
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DOI: 10.1016/J.JMAA.2012.03.001
description: scientific article; zbMATH DE number 6030456