Spectral shift function and eigenvalues of the perturbed operator

In the space of square-integrable functions on the positive semi-axis, two positive selfadjoint operators are constructed that are generated by a one-dimensional free Hamiltonian. These operators are employed to construct a pair of spectrally absolutely continuous bounded selfadjoint operators whose difference is an operator of rank $1$. Then the perturbation determinant is used to find an explicit form of the M. G. Krein spectral shift function for this pair. It is shown that despite the $A$-smoothness of the perturbation in the sense of Hölder, the point $\lambda = 1$, where the spectral shift function has a discontinuity of the first kind, is not an eigenvalue of the perturbed operator.