Renormalized oscillation theory for singular linear Hamiltonian pencils

For many applications, critical information about system dynamics is encoded in associated eigenvalue problems that can be posed as linear Hamiltonian systems with suitable boundary conditions. Motivated by examples from hydrodynamics, quantum mechanics, and magnetohydrodynamics (MHD), we develop a general framework for analyzing a broad class of linear Hamiltonian systems with at least one singular boundary condition and possible nonlinear dependence on the spectral parameter. We 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 of $\mathbb{C}^{2n}$. This extends previous work by the authors for regular linear Hamiltonian systems that depend nonlinearly on the spectral parameter and singular linear Hamiltonian systems that depend linearly on the spectral parameter. We conclude the study by using our framework to study the spectrum in the setting of each of our motivating examples.