A trace formula for canonical differential expressions.
type: Item
instance of: scholarly article
title: A trace formula for canonical differential expressions.
zbMATH Open document ID: 1043.34024
DOI: 10.1016/S0022-1236(02)00083-6
author: Daniel Alpay
author: I. Gohberg
published in: Journal of Functional Analysis
publication date: 2003-04-09T00:00:00Z
review text: The authors study \(2n \times 2n\) canonical differential expressions of the form \[ - i J \frac{d}{dt}f(t,z)=z\,f(t,z) + V(t)\,f(t,z), \quad t \geq 0, \;z \in {\mathbb C}, \] where \(J\) is the block diagonal matrix with entries \(I_n\) and \(-I_n\) and \(V\) is a block anti-diagonal matrix function with entries \(k\) in the right upper and \(k^\ast\) in the left lower corner, \(k\) being an \(n\times n\)-matrix function the entries of which are complex-valued \(L_1( 0,\infty)\) functions. With such a differential expression, two selfadjoint operators \(H_+\) and \(H_-\) are associated, which are formally given by\(H_\pm=-\text{ i} J \frac{d}{dt} \pm V\). It is proved that for every nonreal \(z\) the operator \((H_+-z)^{-1}-(H_--z)^{-1}\) has rank \(n\) and a formula for its trace is derived in terms of the corresponding Weyl function.
reviewed by: Christiane Tretter
Mathematics Subject Classification ID: 34B20
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 47B32
zbMATH DE Number: 1895832
zbMATH Keywords: canonical differential expression
zbMATH Keywords: trace formula
MaRDI profile type: Publication
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full work available at URL: s0022-1236(02)00083-6
OpenAlex ID: W2067370035
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description: scientific article; zbMATH DE number 1895832