Spectral shift function and trace formula |
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Authors: | Kalyan B Sinha and A N Mohapatra |
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Institution: | (1) Indian Statistical Institute, 7, S.J.S. Sansanwal Marg, 110016 New Delhi, India |
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Abstract: | The complete proofs of Krein’s theorem on the spectral shift function and the trace formula are given for a pair of self-adjoint
operators such that either (i) their difference is trace-class or (ii) the difference of their resolvents is trace-class.
The proofs, essentially due to Krein, is based on Herglotz’s theorem on the boundary value of the analytic functions whose
imaginary part is non-negative on the upper half plane, and an almost optimal class of functions are obtained for which the
trace formula is valid. Also an alternative method based on Weyl-von Neumann’s theorem for self-adjoint operators, avoiding
the complex function theory and inspired by Voiculescu’s work, is given for the first case. Furthermore, some applications
of the spectral shift function have been discussed. |
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Keywords: | Spectral shift function trace formula Krein’ s theorem |
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