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On Gelfand-Levitan and Krejn trace formulae. (Russian) Zbl 0751.34040

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)

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.)
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