Local conditions for the existence of the spectral shift function

Let U_o, U₁ be unitary operators in a Hilbert space. If the operator U_x — U_o is nuclear, then (as M. G. Krein established) there exists a function on the unit circle , satisfying the equality for all functions with derivative 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 U_o, U₁ by In this paper (1) and (2) are proved under more general (local) conditions on the pair U_o, U₁. Under these conditions we investigate some properties of the function n and describe the class of functions , which are admissible in (1). Applications to differential operators are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 102–117, 1977.