Singular Bilinear Integrals in Quantum Physics

Mathematics

Brian Jefferies¹ 

[1] School of Mathematics, The University of New South Wales, Sydney, NSW 2052, Australia; E-Mail

关键词: bilinear integration;    tensor products;    operator equations;    double operator integrals;    spectral measure;   

DOI  :  10.3390/math3030563

来源: mdpi

【 摘 要 】

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 equationsfor 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.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland.