A characterization of boundary conditions for regular Sturm-Liouville problems which have the same lowest eigenvalues (Q2477851)

Consider the regular Sturm-Liouville operator defined by \[ \frac{1}{w}\left[ -(py^{\prime })^{\prime}+qy\right] \text{ for }a\leq x\leq b\; \] under the coupled boundary condition \[ Y(a)=e^{-i\theta }KY(b), \] where \(Y(x)=(y(x),\;p(x)y^{\prime }(x))^{t}\) and the matrix \(K\) is unitary, i.e. \(\det K=1.\) For example if \(\theta =0\) and \(K=Id\) then we have the periodic BC, while \(K=-Id\) yields the antiperiodic case. It is known that eigenvalues in these cases can be compared. Here, the authors obtain a similar result, where they compare the eigenvalues in terms of \(\theta \) and \(K.\) For example they show that if \(K_{11}>0\) and \(K_{12}<0\) then \(\nu _{0}\leq \lambda _{0}(K)<\lambda _{0}(e^{i\theta }K)<\lambda _{0}(-K)\leq \left\{ \nu _{1},\;\gamma _{0}\right\} \leq \lambda _{1}(-K)<\) \(\lambda _{1}(e^{i\theta }K)<\dots,\) where \(\nu _{k}\) and \(\gamma _{k}\;\) are respectively the eigenvalues of a Dirichlet \(y(a)=K_{22}y(b)-K_{12}p(b)y^{\prime }(b)=0\) and Neumann type \(p(a)y^{\prime }(a)=K_{21}y(b)-K_{11}p(b)y^{\prime }(b)=0\) boundary conditions.