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

TL;DR

This paper develops a unified spectral theory for complex Hamiltonian systems on Sturmian time scales, introducing Weyl-Sims sets and matrix-valued M-functions, with applications to difference and differential equations.

Contribution

It introduces Weyl-Sims sets and matrix-valued M-functions for non-self-adjoint Hamiltonian systems on Sturmian time scales, unifying discrete and continuous spectral theories.

Findings

Defined Weyl-Sims sets replacing classical Weyl circles

Characterized realizations and constructed resolvents of dynamic operators

Applied theory to even-order scalar and Orr-Sommerfeld equations on 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.