Non-self adjoint Sturm-Liouville problem with spectral and physical parameters in boundary conditions

We present a complete description on the spectrum and eigenfunctions of the following two point boundary value problem $$(p(x)f')'-(q(x)-\lambda r(x))f=0\;, \;\; 0<x<L \quad ; \quad f'(0)=(\alpha_{1} \lambda + \alpha_{2}) f(0) \quad ; \quad f'(L)=(\beta_{1}\lambda -\beta_{2})f(L),$$ where $\lambda$ and $\alpha_{i}, \beta_{i}$ are spectral and physical parameters. Our survey is focused mainly in the case $\alpha_{1}>0$ and $\beta_{1}<0$, where neither self adjoint operator theorems on Hilbert spaces nor Sturm's comparison results can be used directly. We describe the spectrum and the oscillatory results of the eigenfunctions from a geometrical approach, using a function related to the Pr\"ufer angle. The proofs of the asymptotic results of the eigenvalues and separation theorem of the eigenfunctions are developed through classical second order differential equation tools. Finally, the results on the spectrum of the equation are used for the study of the linear instability of a simple model for the fingering phenomenon on the flooding oil recovery process.