Display Abstract

Title Sturmian theory for linear Hamiltonian systems without controllability

Name Roman \v{S}imon Hilscher
Country Czech Rep
Email hilscher@math.muni.cz
Co-Author(s)
Submit Time 2010-02-08 10:13:25
Session
Special Session 48: Differential, Difference, and Dynamic Equations
Contents
Sturmian theory for differential equations is a classical topic in the literature with active research and generalizations in various directions. Classical results of this type, the Sturmian comparison and separation theorems, relate the numbers of zeros or the focal points of two solutions of one or two differential equations. In this talk we present new results in the oscillation theory of linear Hamiltonian systems. In particular, we discuss Sturmian separation and comparison theorems for these systems when no controllability assumption is imposed, which generalizes the traditional results of W.~T.~Reid for controllable linear Hamiltonian systems. Our new theory is based on several recent results by W.~Kratz, M.~Wahrheit, V.~Zeidan, and the author regarding the piecewise constant kernel for conjoined bases of the Hamiltonian system, the oscillation and eigenvalue theorems, and the Rayleigh principle for linear Hamiltonian systems without controllability. References: [1] W.~Kratz, Definiteness of quadratic functionals, {\it Analysis (Munich)}, {\bf 23} (2003), no.~2, 163--183. [2] W.~Kratz, R.~\v{S}imon~Hilscher, Rayleigh principle for linear Hamiltonian systems without controllability, submitted. [3] R.~\v{S}imon~Hilscher, Sturmian theory for linear Hamiltonian systems without controllability, submitted. [4] W.~T.~Reid, {\it Ordinary Differential Equations}, Wiley, New York, 1971.