Titchmarsh–Sims–Weyl theory for complex Hamiltonian systems on Sturmian time scales

Abstract

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.