  We focus on the semi-discrete complex modified Korteweg-de Vries (DcmKdV)
equation in this paper. The direct and inverse scattering theory is developed
with zero and non-zero boundary conditions (BCs) of the potential. For direct
problem, the properties of the eigenfunctions and the scattering matrix,
including analyticity, asymptotics and symmetries, are investigated, which
facilitates the establishment of the Riemann-Hilbert (RH) problems. By solving
the RH problems in the inverse problem part, the reconstruction potential
formulas are obtained, which allows us to derive the N-soliton solutions in the
reflectionless case. Meanwhile, the trace formulas are derived by means of
studying the corresponding RH problems. Furthermore, the dynamic
characteristics of the 1-soliton and 2-soliton with zero and non-zero boundary
are demonstrated by graphical simulation.
