Renormalized oscillation theory for singular linear Hamiltonian systems

Working with a general class of linear Hamiltonian systems on intervals at least one singular endpoint which can be limit-point, limit-circle, or limit-intermediate, we show that renormalized oscillation results obtained in natural way through consideration the Maslov index associated appropriately chosen paths Lagrangian subspaces C2n. In first part analysis associate our families well-defined self-adjoint operators, and latter employ approach to count number eigenvalues these operators have fixed (?1,?2) whose closures do not intersect essential spectrum operators. We conclude two illustrative examples, indicating how theory implemented practice. This extends previous work by authors for regular systems.