Regularized traces of differential operators. The Lidskii-Sadovnichii method (Q2704023)

\textit{V. B. Lidskij} and \textit{V. A. Sadovnickij} [Funkts. Anal. Prilozh. 1, No. 2, 52-59 (1967; Zbl 0176.37001)] devised a method for determining regularized sums of zeros of functions belonging to a class \(K\) of entire functions: for \(f\in K\), such a sum is of the form \(\sum_n[z_n^m-A_m(n)]\), \(m\in\mathbb{N}\), where \(z_n\) are the zeros of \(f(z)\) and \(A_m(n)\) are numbers which are given by the asymptotics of \(z_n\) and which ensure the convergence of the series. The asymptotic structure of functions in \(K\) is described by the asymptotics of the principal solution system of a linear \(n\)th-order differential equation on \([0,1]\), whose coefficients are polynomials in a spectral parameter \(\lambda\). The asymptotics of the solutions to this equation as \(\lambda\to\infty\) depends on the roots of the associated Tamarkin polynomial \(\prod(\omega)\).NEWLINENEWLINENEWLINEThe author applies the Lidskij-Sadovnichij technique to cases in which \(\prod(\omega)\) has multiple zeros.