Spectral theory -- 50 years of progress and a conclusion (Q2725065)

The title of this interesting paper is somewhat misleading. One should not expect in reading it a description of the development of spectral theory in the first-half of the 20th century.NEWLINENEWLINENEWLINEThe paper indeed gives a clear and readable survey of the development of the spectral analysis of second-order linear ordinary differential operators acting in the Hilbert space \(L^2(0,\infty)\). After distinguishing the limit point and the limit circle cases introduced by Weyl, the author explains the concepts of spectral measure and of the Weyl-Titshmarsh \(m\)-function associated to such operators. He analyzes the singular and absolutely continuous spectra, states new results about the location of the absolutely continuous spectrum, and concludes by a short description of recent results of Kiselev, which show that important developments can still be expected in this classs of problems.