A note on the equivalence between even order Sturm-Liouville equations and symplectic systems on time scales. (English) Zbl 1348.34146

Zemánek, Petr

Appl. Math. Lett. 26, No. 1, 134-139 (2013).

Summary: The \(2n\)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 \(2n\)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.

MSC:

34N05	Dynamic equations on time scales or measure chains
34B24	Sturm-Liouville theory
34B20	Weyl theory and its generalizations for ordinary differential equations
39A12	Discrete version of topics in analysis

Keywords:

time scale; even order Sturm-Liouville dynamic equation; time reversed symplectic system; quadratic functional