Spectral problem for a triple differentiation operator with asymmetric weight (Q2399582)

Consider the boundary value problem \[ \begin{gathered} y'''(x)= \lambda\rho(x) y(x)\quad\text{for }0<x<\pi,\\ U_j[y]= y^{(j)}(0)-(-1)^j\,\alpha y^{(j)}(\pi)= 0\quad\text{for }j= 0,1,2,\end{gathered}\tag{\(*\)} \] where \(\alpha\) is a nonzero real number and \(\rho\) is an integrable complex-valued function satisfying \[ \rho(x)+ \rho(\pi-x)\equiv 0\quad\text{for }0<x<\pi/2. \] The authors prove that in the case that \(\alpha\) takes the value \(1\) in the boundary conditions \(U_0\) and \(U_2\), the spectrum of \((*)\) fills the entire complex plane.