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

2010/01/22 by Douglas R. Anderson, Anderson, Douglas R.
Mathematics · Physics and Astronomy · #34B20 #34B27 #34N05 #47A10 #47B39 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.CA #msc:34B20 #msc:34B27 #msc:34N05 #msc:47A10 #msc:47B39

paper · pdf · doi:10.48550/arxiv.1001.3912

27 pages. Note that the results for even-order equations are new for T=R and T=Z as well as on Sturmian times scales generally

arxiv created 2010/01/22 · openalex publication_date 2010/01/22 · arxiv updated 2010/02/26 · openalex created_date 2022/10/03 · openalex updated_date 2026/08/04

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.

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