Spectral analysis of the weighted Laplacian in slip and no-slip flows

Slip boundary conditions for the velocity field impact on the spectral properties of the advection-diffusion operator describing transport of passive particles in laminar parallel flows. By considering the Hermitian operator (referred to as the weighted Laplacian), describing the interplay between axial convection and cross-sectional diffusion of a scalar field, we show that the spectral watershed between slip and no-slip boundary conditions is a qualitatively different scaling behavior of the mean of the normalized eigenfunctions of the weighted Laplacian. The occurrence of slip conditions also influences the scaling of the density of states as regards both the leading and the subleading term in the Weyl's expansion.