  We study non-self-adjoint Hamiltonian systems on Sturmian time scales,
defining Weyl-Sims sets, which replace the classical Weyl circles, and a
matrix-valued $M-$function on suitable cone-shaped domains in the complex
plane. Furthermore, we characterize realizations of the corresponding dynamic
operator and its adjoint, and construct their resolvents. Even-order scalar
equations and the Orr-Sommerfeld equation on time scales are given as examples
illustrating the theory, which are new even for difference equations. These
results unify previous discrete and continuous theories to dynamic equations on
Sturmian time scales.
