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Local conditions for the existence of the spectral shift function

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Let Uo, U1 be unitary operators in a Hilbert space. If the operator Ux — Uo is nuclear, then (as M. G. Krein established) there exists a function Π on the unit circle\(\mathbb{T}, \eta = \eta \left( {\mathcal{U}_1 , \mathcal{U}_0 } \right) \eta \in L^1 \left( \mathbb{T} \right)\), satisfying the equality

for all functions ϕ with derivative ϕ′ from the Wiener class. M. Sh. Rirman and M. G. Krein proved that the function n is connected with the scattering matrix S for the pair Uo, U1 by

In this paper (1) and (2) are proved under more general (local) conditions on the pair Uo, U1. Under these conditions we investigate some properties of the function n and describe the class of functions η, which are admissible in (1). Applications to differential operators are given.

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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 102–117, 1977.

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Koplienko, L.S. Local conditions for the existence of the spectral shift function. J Math Sci 34, 2080–2090 (1986). https://doi.org/10.1007/BF01741582

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  • Hilbert Space
  • Local Condition
  • Differential Operator
  • Unit Circle
  • Unitary Operator