Singular bilinear integrals in quantum physics
type: Item
instance of: scholarly article
title: Singular bilinear integrals in quantum physics
zbMATH Open document ID: 1330.81093
published in: Mathematics
publication date: 2016-01-12T00:00:00Z
review text: Summary: Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how ``decoupled'' bilinear integration may be used to find solutions \(X\) of operator equations \(AX-XB=Y\) with respect to the spectral measure of \(A\) and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space \(L^1((P\otimes Q)_{\mathcal L(\mathcal H)})\) of double operator integrable functions for spectral measures \(P\), \(Q\) acting in a Hilbert space \(\mathcal H\) and applied to the representation of the trace of \(\int_{\Lambda\times\Lambda}\varphi\) \(d(PTP)\) for a trace class operator \(T\). The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function.
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81U05
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 6528839
zbMATH Keywords: bilinear integration
zbMATH Keywords: tensor products
zbMATH Keywords: operator equations
zbMATH Keywords: double operator integrals
zbMATH Keywords: spectral measure
author: Brian Jefferies
MaRDI profile type: Publication
full work available at URL: math3030563
OpenAlex ID: W1532045997
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DOI: 10.3390/MATH3030563
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description: scientific article; zbMATH DE number 6528839