Completeness of eigenfunctions of one class of pencils of differential operators with constant coefficients
Consider in the space \(L_2[0,1]\) the pencil \(L(\lambda)\) of ordinary differential operators generated by the differential expression \(l(y,\lambda):=\sum_{s+k=n}p_{sk}\lambda^sy^{(k)}\), \(p_{sk}\in\mathbb C\), \(p_{0n}\neq0\), and linearly independent homogeneous boundary conditions \[ \begin{aligned} U_j(y,\lambda)&=\sum_{s+k=\kappa_j}\lambda^s\alpha_{jsk}y^{(k)}(0)=0,\quad j=\overline{1,n-1},\\ U_n(y,\lambda)&=\sum_{s+k=\kappa_n}\lambda^s\left(\alpha_{nsk}y^{(k)}(0)+\beta_{nsk}y^{(k)}(1)\right)=0, \end{aligned} \] where \(\lambda\in\mathbb C\) is a spectral parameter, \(\alpha_{jsk},\beta_{nsk}\in\mathbb C\). Let the roots of the characteristic equation \(\sum_{s+k=n}p_{sk}\omega^k=0\) be nonzero and distinct; assume that they belong to one and the same ray that goes out of the origin. The author gives sufficient conditions for the simple completeness of the system of eigen- and associated functions of the pencil \(L(\lambda)\) in spaces \(L_2[0,1]\) and \(L_2[0,\sigma]\) for appropriate \(0<\sigma<1\).
simple and multiple completeness of the system of eigenfunctions
scientific article; zbMATH DE number 5621428