On an eigenvalue problem arising in the study of the stability of ocean currents

We study a singular Sturm-Liouville eigenvalue problem which arises in studying the stability of rotating shallow-water shear flows on an equatorial -plane. Earlier studies have shown that the problem has only stable eigenvalues if a certain parameter exceeds 3/4. On the other hand, numerical studies have found unstable eigenvalues if is sufficiently small. In this note, we give an analytical proof that the bound of 3/4 is indeed sharp, i.e. there are instabilities for any < 3/4.