In this paper we derive new singular Sturmian separation theorems for nonoscillatory symplectic difference systems on unbounded intervals. The novelty of the presented theory resides in two aspects. We introduce the multiplicity of a focal point at infinity for conjoined bases, which we incorporate into our new singular Sturmian separation theorems. At the same time we do not impose any controllability assumption on the symplectic system. The presented results naturally extend and complete the known Sturmian separation theorems on bounded intervals by J. Elyseeva (2009), as well as the singular Sturmian separation theorems for eventually controllable symplectic systems on unbounded intervals by O. Dosly and J. Elyseeva (2014). Our approach is based on developing the theory of comparative index on unbounded intervals and on the recent theory of recessive and dominant solutions at infinity for possibly uncontrollable symplectic systems by the authors (2015 and 2017). Some of our results, including the notion of the multiplicity of a focal point at infinity, are new even for an eventually controllable symplectic difference system.

Last change: 27.10.2018. (c) Roman Simon Hilscher