Lyubishkin, V. A. On Gelfand-Levitan and Krejn trace formulae. (Russian) Zbl 0751.34040 Mat. Sb. 182, No. 12, 1786-1795 (1991). The trace formula \(\sum_ n[\mu_ n-\lambda_ n-(P\varphi_ n,\varphi_ n)]=0\) (a generalization of Gel’fand-Levitan and Krejn trace formulae) is considered. Here \(\{\lambda_ n\}\) and \(\{\varphi_ n\}\) are discrete spectrum and eigenfunctions of a semi-bounded from below, self-adjoint operator; and \(P\) is a bounded, self-adjoint operator with discrete spectrum \(\{\mu_ n\}\). The upper formula is proved when (difficult to check) Tauber-type conditions are satisfied. The absolute convergence of the upper sum is proved when stronger conditions on the distribution of the eigenvalues \(\{\lambda_ u\}\) are satisfied. Reviewer: Y.P.Mishev (Sofia) Cited in 1 Review MSC: 34L05 General spectral theory of ordinary differential operators 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) Keywords:trace formula; Gel’fand-Levitan and Krejn trace formulae; discrete spectrum; eigenfunctions; Tauber-type conditions; absolute convergence PDFBibTeX XMLCite \textit{V. A. Lyubishkin}, Mat. Sb. 182, No. 12, 1786--1795 (1991; Zbl 0751.34040) Full Text: EuDML