The asymptotics of eigenvalues and eigenfunctions and a trace formula for a potential with Delta functions
Het, the authors extend two results on the boundary problem \[ z(x)=x-\int_{0}^{l}{\nu(x-t)z(t)}\,d(\sigma(t)+t\lambda), \quad x\in{[0,l]}, \quad z(l)=0, \] obtained by them in two previous papers [\textit{V. A.Vinokurov}, Dokl. Math., 57, No. 1, 43--46 (1998; Zbl 0989.34016) and Dokl. Math., 59, No. 2, 220--222 (1999; Zbl 0967.34076)]. In these papers, they considered the case of an absolutely continuous function \(\sigma\). In the paper under review, functions \(\sigma\) having \(\delta\)-type discontinuities are allowed. Namely, one assumes that \(\sigma\) is in the space \(BV_{c}[0,l]\) of functions with bounded variation, right continuous on \([0,l]\) and continuous at \(0\) and \(l\). By approximating \(\sigma\) with smooth functions and passing to the limit one proves that the second-order approximate formulas for the spectral parameter \(\lambda_{n}\) and for the eigenfunctions which were obtained in the first paper cited above are still valid. The regularized trace formula from the second paper cited above is also extended to the case of functions from \(BV_{c}[0,l]\).
asymptotics of eigenvalues and eigenfunctions
scientific article; zbMATH DE number 1916101