Spectral Shift Function and Eigenvalues of the Perturbed Operator

In the space of square integrable functions on the positive semi-axis, two positive self-adjoint operators generated by a one-dimensional free Hamiltonian are constructed. These operators are employed to construct a pair of spectrally absolutely continuous bounded self-adjoint operators whose difference is an operator of rank 1. 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 Asmoothness of the perturbation in the sense of Hölder, the point λ = 1, where the spectral shift function has a discontinuity of the first kind, is not an eigenvalue of the perturbed operator.