Spectral properties of boundary value problems for functional-differential equations with integrable coefficients
The author considers boundary value problems for a functional-differential equation of the form \[ -y^{\prime\prime}(x)+q(x)y(x)=\lambda a^2y(x)+\alpha y(b),\tag{1} \] \[ 0\leq x\leq \pi,\qquad 0<b<\pi,\qquad \alpha>0, \] where \(\alpha\in\mathbb{R}\) and \(\lambda\) is a spectral parameter, with boundary conditions of the following form: \[ y(0)=y(\pi)=0; \] \[ y^\prime(0)=hy(0);\quad y^\prime(\pi)=-Hy(\pi);\quad h,H\in\mathbb{R}; \] \[ y(0)=0;\quad y^\prime(\pi)=0; \] \[ y^\prime(0)=0;\quad y(\pi)=0. \] The potential \(q(x)\) belongs to \(\mathcal{L}_1[0,\pi]\), or, which is the same, satisfies \[ \left(\int_0^xq(t)dt\right)_x^\prime=q(x) \] for almost all \(x\in[0,\pi]\). The author extends the method for constructing asymptotic formulas of arbitrary-order of accuracy for the eigenvalues and eigenfunctions developed by \textit{V. A. Vinokurov} and \textit{V. A. Sadovnichij} [Izv. Math. 64, No.~4, 695--754 (2000); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No.~4, 47--108 (2000; Zbl 1001.34021)] to equation (1).
scientific article; zbMATH DE number 5820357