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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. [ABSTRACT FROM AUTHOR]

Abstract: Consider a unitary operator acting on a complex separable Hilbert space . In this paper we study spectral properties for perturbations of of the type, with K a compact self-adjoint operator acting on and β a real parameter. We apply the commutator theory developed for unitary operators in Astaburuaga et al. (2006) to prove the absence of singular continuous spectrum for . Moreover, we study the eigenvalue problem for when the unperturbed operator does not have any. A typical example of this situation corresponds to the case when is purely absolutely continuous. Conditions on the eigenvalues of K are given to produce eigenvalues for for both cases finite and infinite rank of K, and we give an example where the results can be applied. [Copyright &y& Elsevier]

Abstract: In this paper we investigate perturbations of the regular spectrum of an upper triangular operator matrix such as acting on a Hilbert space . [Copyright &y& Elsevier]