On \(A\)-integrability of the spectral shift function of unitary operators arising in the Lax-Phillips scattering theory (Q1918241)

For a pair of unitary operators \((U,U_1)\) arising in the Lax-Phillips scattering theory such that \(U-U_1\) is a Hilbert-Schmidt operator the following trace formula \[ V.p. \;tr\{\phi (U) - \phi (U_1)\} = (A) \int_{|\xi|=1}\eta(\xi)d\phi(\xi) \] is proved, where in the right hand side we have an \(A\)-integral (in Kolmogorov-Titchmarsh's sense). Necessary and sufficient conditions for the Lebesgue integrability of \(\eta\) are found.