On a Sturm-Liouville problem with spectral and physical parameters in boundary conditions (Q2796891)

Consider the two-point boundary value problem NEWLINE\[NEWLINE\begin{aligned} &(p(x)f')'-(q(x)-\lambda r(x))f=0, \;0<x<L,\\ & f'(0)=(\alpha _1\lambda +\alpha _2)f(0), \;f'(L)=(\beta _1\lambda -\beta _2)f(L), \end{aligned}NEWLINE\]NEWLINE where \(p\), \(q\), \(r\) are positive and regular functions on \([0,L]\), \(\alpha _i\), \(\beta _i\) are real and \(\lambda \) is the spectral parameter. It is assumed that \(\alpha _1>0\) and \(\beta _1<0\). Using a Darboux-Crum transformation, the eigenvalue problem is transformed into a self-adjoint problem with \(\lambda \)-independent boundary conditions. Therefore, the eigenvalues are real (and simple). It is shown that the eigenvalues can be indexed as \(\lambda _{-1}<\lambda _{-0}<0< \lambda _{0}<\lambda _1<\lambda _2\cdots\) and that the eigenfunctions corresponding to \(\lambda _l\) have \(|l|\) zeros in \((0,L)\). Further results are a separation theorem for zeros of the solution of the differential equation satisfying the boundary condition at \(0\) and the formula \(\sqrt{\lambda _n}=n\pi /L +O(n^{-1})\) as \(n\to \infty \). As an application, linear stability in three layer Hale-Shaw flows is investigated.