Discrete point spectrum of linear stability operator for Saffman–Taylor bubbles with nonzero surface tension

We will study the linear spectral properties of steady Saffman–Taylor bubbles with surface tension in a Hele-Shaw cell. It was known [21] that the Saffman–Taylor exact solution without surface tension is linearly unstable and the point spectrum is a continuum set. In this paper, we rigorously conclude that the point spectrum of the stability operator of any branch of Saffman–Taylor bubbles is a discrete set when surface tension is not zero.