A note on the equivalence between even order Sturm–Liouville equations and symplectic systems on time scales

Abstract

The 2 n th -order Sturm–Liouville differential and difference equations can be written as linear Hamiltonian differential systems and symplectic difference systems, respectively. In this work, a similar result is given for the 2 n th -order Sturm–Liouville equation on time scales using time reversed symplectic dynamic systems. Moreover, we show that this transformation preserves the value of the corresponding quadratic functionals which enables a further generalization of the theory for continuous and discrete Sturm–Liouville equations.