The linear instability study of the viscous Rayleigh-Taylor model in the neighborhood of a laminar smooth increasing density profile ρ 0 (x 3) amounts to the study of the following ordinary differential equation of order 4 − λ 2 [ρ 0 k 2 φ − (ρ 0 φ) ] = λμ(φ (4) − 2k 2 φ + k 4 φ) − gk 2 ρ 0 φ, (0.1) where λ is the growth rate in time, and k is the wave number transverse to the density profile. In the case of ρ 0 ≥ 0 compactly supported, we provide a spectral analysis showing that in accordance with the results of Helffer and Lafitte (Asymptot Anal 33:189-235, 2003), there is an infinite sequence of non-trivial solutions (λ n , φ n) of (0.1), with λ n → 0 when n → +∞ and φ n ∈ H 4 (R). In the more general case where ρ 0 > 0 everywhere and ρ 0 converges at ±∞ to finite limits ρ ± > 0, we prove that there exist finitely non-trivial solutions (λ n , φ n) of (0.1). The line of investigation is to reduce both cases to the study of a self-adjoint operator on a compact set. Keywords Rayleigh-Taylor instability • Linear growth rate • Self-adjoint operator • Spectral theory Mathematics Subject Classification 34B05 • 47A05 • 47A55 • 47B07 • 76D05 B Ti én-Tài Nguy ẽn