  In this paper we study numerical positivity and contractivity in the infinite
norm of Crank-Nicolson method when it is applied to the diffusion equation with
homogeneous Dirichlet boundary conditions. For this purpose, the amplification
matrices are written in terms of three kinds of Chebyshev-like polynomials, and
necessary and sufficient bounds to preserve the desired qualitative properties
are obtained. For each spatial mesh, we provide the equations that must be
solved as well as the intervals that contain these bounds; consequently, they
can be easily obtained by a bisection process. Besides, differences between
numerical positivity and contractivity are highlighted. This problem has also
been addressed by some other authors in the literature and some known results
are recovered in our study. Our approach gives a new insight on the problem
that completes the panorama and that can be used to study qualitative
properties for other problems.
