On the basis properties of root functions of two generalized eigenvalue problems (Q425247)

Consider the boundary eigenvalue problems \[ \begin{gathered} -u''(x)=\lambda u(-x)\quad\text{for }-1<x<1,\\ u'(-1)=\alpha u'(1),\quad u(-1)= u(1),\end{gathered}\tag{1} \] and \[ \begin{gathered} -u''(x)=\lambda u(-x)\quad\text{for }-1< x< 1,\\ u'(-1)= u'(1)- u(-1)- u(1),\quad u(-1)+\alpha u(1)= 0.\end{gathered}\tag{2} \] The authors prove that in the case \(\alpha^2\neq 1\), the system of root functions of the boundary eigenvalue problems (1) and (2) form a Riesz basis in \(L_2(-1,1)\).