  This paper studies the stability of weak dispersive shock profiles for a
quantum hydrodynamics system in one space dimension with nonlinear viscosity
and dispersive (quantum) effects due to a Bohm potential. It is shown that, if
the shock amplitude is sufficiently small, then the profiles are spectrally
stable. This analytical result is consistent with numerical estimations of the
location of the spectrum (Lattanzio, Zhelyazov, Math. Models Methods Appl. Sci.
31, 2021). The proof is based on energy estimates at the spectral level, on the
choice of an appropriate weighted energy function for the perturbations
involving both the dispersive potential and the nonlinear viscosity, and on the
montonicity of the dispersive profiles in the small-amplitude regime.
