Renormalized oscillation theory for singular linear Hamiltonian systems (Q2139179)

The authors consider a general class of linear Hamiltonian systems on intervals with at least one singular endpoint; this can be a limit point, a limit circle, or limit-intermediate. They show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces. The first part of the work is devoted to associating to the linear Hamiltonian systems in the considered class a family of well-defined self-adjoint operators. After this, the authors employ the renormalized oscillation approach to count the number of eigenvalues these operators have on fixed intervals whose closures do not intersect the essential spectrum of the operators. The analysis is completed by two illustrative examples, which show how the theory can be implemented in practice. This paper extends previous work by the authors, dealing with regular linear Hamiltonian systems.