Generalized reciprocity principle for discrete symplectic systems

Abstract

This paper studies transformations for conjoined bases of symplectic difference systems $Y_{i+1}=\mathcal S_{i}Y_{i}$ with the symplectic coefficient matrices $\mathcal S_i.$ For an arbitrary symplectic transformation matrix $P_{i}$ we formulate most general sufficient conditions for $\mathcal S_{i},\, P_{i}$ which guarantee that $P_{i}$ preserves oscillatory properties of conjoined bases $Y_{i}.$ We present examples which show that our new results extend the applicability of the discrete transformation theory.

Keywords

• symplectic difference systems
• focal points
• symplectic transformations
• reciprocity principle
• comparative index