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Mathematics 2015, 3(3), 563-603; doi:10.3390/math3030563

Singular Bilinear Integrals in Quantum Physics

School of Mathematics, The University of New South Wales, Sydney, NSW 2052, Australia
Academic Editor: Palle E.T. Jorgensen
Received: 6 April 2015 / Revised: 12 June 2015 / Accepted: 17 June 2015 / Published: 29 June 2015
(This article belongs to the Special Issue Mathematical physics)
View Full-Text   |   Download PDF [475 KB, uploaded 29 June 2015]

Abstract

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. View Full-Text
Keywords: bilinear integration; tensor products; operator equations; double operator integrals; spectral measure bilinear integration; tensor products; operator equations; double operator integrals; spectral measure
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Jefferies, B. Singular Bilinear Integrals in Quantum Physics. Mathematics 2015, 3, 563-603.

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