On October 11, 2023, the next All-Institute seminar was held at the AR ETN Institute of Mathematics and Mechanics. In the all-institute seminar of IMM, chief researcher employee of the department “Functional analysis”, doctor of mathematical sciences, prof. Elshad Hatam oglu Eyvazov gave a report on  the topic “Spectral shift function and eigenvalues of the perturbed operator”.

The report will in the space of square-integrable functions on the positive semi-axis, two positive self-adjoint 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 self-adjoint 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 Holder, the point where the spectral shift function has a discontinuity of the first kind, is not an eigenvalue of the perturbed operator.

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