Positivity of quadratic functionals on time scales: Necessity

In this work we establish that disconjugacy of a linear Hamiltonian system on time scales is a necessary condition for the positivity of the correspon ding quadratic functional. We employ a certain minimal normality (controlla bility) assumption. Pence, the open problems stated by the author in [17], [18] are solved with the result that positivity of the quadratic functional is equivalent to disconjugacy of the Hamiltonian system on the interval un der consideration. The general approach on time scales T involves, as speci al cases, the well-known continuous case for T = R and recently developed d iscrete one for T = Z, so that they are unified. As applications, Sturmian type separation and comparison theorems on time scales are also provided.