On completeness of eigenfunctions of quadratic bundles of ordinary differential operators (Q1311191)

The differential equation (1) \(y^ 2-2 \alpha \lambda y'+b \lambda^ 2 y=0\) with boundary conditions (2) \(\alpha_{11} y'(0)+\lambda \alpha_{10} y(0)=0\), \(\beta_{21} y'(1)+\lambda \beta_{20} y(1)=0\) is considered, where the coefficients are constant. It is assumed that the characteristic equation \(\omega^ 2-2a \omega+b=0\) of (1) has two distinct nonzero roots \(\omega_ 1\), \(\omega_ 2\) lying on a common ray emanating from the origin with \(\tau=| \omega_ 2 |/ | \omega_ 1 |>1\). The aim of the paper is to obtain statements on completeness and minimality of the system \(Y\) of eigenvectors, possibly augmented by a constant function. Some results proved are: For each \(0<\sigma \leq 1\) the system \(Y\) is not double complete in \(L_ 2 [0,\sigma]\) (Theorem 1); the system \(Y\) is complete in \(L_ 2 [0,\sigma]\) if a certain linear homogeneous functional equation has only the trivial solution (Theorem 2) and if \(0<\sigma \leq 1-1/\tau\) (Theorem 4); the system \(Y\) is minimal in \(L_ 2 [0,\sigma]\) if a corresponding linear nonhomogeneous functional equation has a solution in \(L_ 2 [0,\sigma]\) (Theorem 3); the system \(Y\) is not complete in \(L_ 2 [0,\sigma]\) with infinite defect if \(\sigma>1-1/ \tau\) and \(\tau<| a_ 1/a_ 2 |^ 2\), where \(a_ i=\alpha_{11} \omega_ i+\alpha_{10}\) (Theorem 6).