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Working with a general class of regular linear Hamiltonian systems, 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 C 2 n . We verify that our applicability class includes Dirac and Sturm–Liouville systems, as well as a system arising from differential-algebraic equations for which the spectral parameter appears nonlinearly. [ABSTRACT FROM AUTHOR]

In this paper we present the theory of oscillation numbers and dual oscillation numbers for continuous Lagrangian paths in R 2 n . Our main results include a connection of the oscillation numbers of the given Lagrangian path with the Lidskii angles of a special symplectic orthogonal matrix. We also present Sturmian type comparison and separation theorems for the difference of the oscillation numbers of two continuous Lagrangian paths. These results, as well as the definition of the oscillation number itself, are based on the comparative index theory (Elyseeva, 2009). The applications of these results are directed to the theory of Maslov index of two continuous Lagrangian paths. We derive a formula for the Maslov index via the Lidskii angles of a special symplectic orthogonal matrix, and hence we express the Maslov index as the oscillation number of a certain transformed Lagrangian path. The results and methods are based on a generalization of the recently introduced oscillation numbers and dual oscillation numbers for conjoined bases of linear Hamiltonian systems (Elyseeva, 2019 and 2020) and on the connection between the comparative index and Lidskii angles of symplectic matrices (Šepitka and Šimon Hilscher, 2021). [ABSTRACT FROM AUTHOR]

In the previous papers (Hu et al. in J Differ Equ 266(7):4106–4136, 2019; Kong et al. in J Differ Equ 156(2):328–354, 1999), the jump phenomena of the j-th eigenvalue were completely characterized for Sturm–Liouville problems. In this paper, we show that the jump number of these eigenvalue branches is exactly the Maslov index for the path of corresponding boundary conditions. Then we determine the sharp range of the jth eigenvalue on each layer of the space of boundary conditions. Finally, we prove that the graph of monodromy matrix tends to the Dirichlet boundary condition as the spectral parameter goes to - ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we prove Morse index theorem of Lagrangian systems with self-adjoint boundary conditions. Based on it, we give some nontrivial estimates on the difference of Morse indices. As an application, we get a new criterion for the stability problem of brake periodic orbit. [ABSTRACT FROM AUTHOR]

Working with general linear Hamiltonian systems on [0, 1], and with a wide range of self-adjoint boundary conditions, including both separated and coupled, we develop a general framework for relating the Maslov index to spectral counts. Our approach is illustrated with applications to Schrödinger systems on R with periodic coefficients, and to Euler-Bernoulli systems in the same context. [ABSTRACT FROM AUTHOR]

We study the Maslov index as a tool to analyze stability of steady state solutions to a reaction-diffusion equation in one spatial dimension. We show that the path of unstable subspaces associated to this equation is governed by a matrix Riccati equation whose solution S develops singularities when changes in the Maslov index occur. Our main result proves that at these singularities the change in Maslov index equals the number of eigenvalues of S that increase to $$+\infty $$ minus the number of eigenvalues that decrease to $$-\infty $$ . [ABSTRACT FROM AUTHOR]