Basis properties of eigenfunctions of second-order differential operators with involution (Q1925394)

Consider the spectral problem \[ -u''(-x)+\alpha u''(x)=\lambda u(x),\tag{\(*\)} \] where \(\lambda\) is the spectral parameter. The authors prove the following results. Theorem 1. Suppose that \(\alpha^2\neq 1\). Then the eigenfunctions of \((*)\) satisfying \[ u(-1)= u(1) =0 \] form an orthonormal basis in \(L_2(-1,1)\). Theorem 2. Assume \(\alpha^2\neq 1\). Then the eigenfunctions of \((*)\) satisfying \[ u(-1)= u(1),\;u'(-1)= u'(1)\qquad(\text{periodic case}) \] or \[ u(-1)= -u(1),\;u'(-1)= -u'(1)\qquad(\text{antiperiodic case}) \] form orthonormal basis in \(L_2(-1,1)\).