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Abstract

Mikhail Shlemovich Birman reached the age of 75 on the 17th January, 2003. An eminent mathematician, he is the author of numerous fundamental results in the general spectral theory and in the spectral theory of differential operators, both ordinary and partial. He contributed much to several other fields of analysis: function space theory, approximation theory, integral operators, etc.A detailed survey of M. B.'s achievements before 1998 can be found in [1 * ] (see also [3 * ]). For this reason, in the present note we restrict ourselves to the description of his results obtained after 1998, referring to earlier publications only when necessary.Nowadays M. B. is as active as ever. Largely, his new results pertain to two fields: 1) the discrete spectrum of a perturbed operator in gaps of the unperturbed one, and 2) absolute continuity of the spectrum of periodic operators of mathematical physics. Before analyzing these results, we would like to emphasize a characteristic feature of M. B.'s entire work: he knows how to look at a specific problem from a bird's-eye view. Rising over individual peculiarities of a concrete question, he states a new and general problem that includes the initial one (and many others) as a particular case. Next he analyzes the general problem in abstract terms. This either automatically leads to a solution of the initial problem, or-in more complicated cases-reveals questions of analytic nature to be answered for that.The bibliography completing this text consists of two parts. In the first part we give references to the papers [1 * , 3 * ] mentioned above and to the list [2 * ] comprising the publications by M. B. over the same period of time. Five publications occurring in [2 * ] are also included in the first part, with the same numbers as in [2 * ]. For various reasons, the corresponding bibliographic data in [2 * ] were not quite complete, and now we give full references. The second part is the full list of subsequent publications by M. B. When making a reference to the first part, we mark the corresponding number with * . Also, in the text the reader may see references of the type [110 * ] that are absent in both parts of the list. Such a reference means the paper number 110 in the earlier list [2 * ]. §1. Spectrum in gaps. Threshold effects Let A be a selfadjoint operator with a gap (λ − , λ + ) in the spectrum. This means that the interval (λ − , λ + ) is free of points of the spectrum, and the edges λ − and λ + belong to the spectrum (or are infinite).Let V be another operator (a "perturbation"); for definiteness, we assume that V ≥ 0. Denote A ± (α) = A ∓ αV . If V is subordinate to A in an appropriate sense, then for any α > 0, the spectrum of A ± (α) in the gap of A either is empty, or consists of at most countably many eigenvalues (with regard to multiplicity), with the only possible accumulation point λ ± . Moreover, as α increases, the eigenvalues of the operator A + (α) (of A − (α)) move leftwards (rightwards).