Central force and the higher rank numerical range
type: Item
instance of: scholarly article
title: Central force and the higher rank numerical range
zbMATH Open document ID: 1238.70007
author: Mao-Ting Chien
author: Hiroshi Nakazato
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-02-27T00:00:00Z
review text: Let \(r=r(\theta)\) be the orbit of a point mass under a central force \(f(r)=-1/r^3\) with angular momentum \(M\). Let \(p=M/\sqrt{M^2-1}\). It is shown that the orbit is a transcendental curve if \(p\) is irrational, and that the orbit is an algebraic curve \(F_A(1,x,y)=0\) for some \(2m \times 2m\) nilpotent Toeplitz matrix \(A\) if \(p=m/j_0\) is rational, where \[ F(t,x,y) = F_A(t,x,y)=\det(t I_n + x (A+A^{\ast})/2+y (A-A^{\ast})/(2 i)). \] Furthermore, the rank-\(k\) numerical range \(\Lambda_k(A)\) of this nilpotent Toeplitz matrix is examined, showing that he sum of numbers of flat portions on the boundary of \(\Lambda_k(A)\), \(k=1,\cdots,m\) is equal to \((m-1 (2 m-3))\).
reviewed by: Ludwig Kohaupt
Mathematics Subject Classification ID: 70F15
zbMATH DE Number: 6009679
zbMATH Keywords: algebraic curve
zbMATH Keywords: Toeplitz matrix
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.12.009
OpenAlex ID: W2068758139
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DOI: 10.1016/J.JMAA.2011.12.009
description: scientific article; zbMATH DE number 6009679