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

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