Abstract
We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale𝕋, which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for𝕋=ℝand𝕋=ℤwithin one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i)M(λ)theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.
Highlights
Singular spectral problems of second-order self-adjoint scalar difference equations over infinite intervals were first studied by Atkinson 1
His work was followed by Agarwal et al 29, Bohner, Bohner et al, Clark and Gesztesy, Shi 33, and Sun et al
The theory of time scales was introduced by Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analysis 36
Summary
The study of spectral problems for differential operators has played an important role in theoretical and in practical aspects. The Weyl-Titchmarsh theory is an important milestone in the study of spectral problems for linear ordinary differential equations It has started with the celebrated work by H. He gave a dichotomy of the limit-point and limit-circle cases for singular spectral problems of secondorder formally self-adjoint linear differential equations. He was followed by Titchmarsh 12 and many others. Further study continued in the 1960s and 1970s with the work of Atkinson 1 on regular Hamiltonian systems xtAtxtBt λW2 t u t , 1.1 u t C t − λW1 t x t − A∗ t u t and Everitt and Kumar 15, 16 on higher-order scalar problems. In the 2000s, Brown and Evans , Clark and Gesztesy , Qi and Chen , Qi 25 , Remling 21 , Shi , Sun et al , Zheng and Chen have made progress by considering spectral problems for Hamiltonian differential systems
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