Evaluating the weight \(\zeta\)-function of the Sturm-Liouville operator (Q804783)

For the operator \((1)\quad \ell (y)=-y''+q(x)y,\) \((2)\quad y'(0)- hy(0)=0,\quad y'(\pi)+Hy(\pi)=0,\) where \(q(x)\in C^{\infty}[0,\pi]\), h, H are real numbers, we propose a new method of calculating the values and the residues of its weight \(\zeta\)-function \(Z(s)=\sum (\sqrt{\lambda_ n})^{-s}/\alpha_ n\) (here \(\lambda_ n\) are eigenvalues of operator (1)-(2), \(\alpha_ p\) are its valuation numbers, and summation is done over all \(n\geq 0)\) at integer nonpositive points. The weight \(\zeta\)- function of the Sturm-Liouville operator has been studied at length by \textit{V. A. Sadovnichij} [Differ. Uravn. 10, No.10, 1808-1818 (1974; Zbl 0311.34028)]. A method was proposed for calculating the values of this function which utilized an asymptotic expansion of the operator's eigenfunctions at large values of the spectral parameter. The method proposed in the present paper can calculate the numbers Z(-2k),\(_{s=- \ell -2k}Z(s)\), \(k=0,1,...\), directly on the basis of the potential q(x) and the coefficient h. Also, in the case where q(x) is analytic, we prove the uniqueness theorem for the reconstruction of q(x) and h from values and residues of the weight \(\zeta\)-function.