Oscillation numbers for continuous Lagrangian paths and Maslov index

Abstract

In this paper we present the theory of oscillation numbers and dual oscillation numbers for continuous Lagrangian paths in R 2n . 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 and on the connection between the comparative index and Lidskii angles of symplectic matrices ( Šepitka and Šimon Hilscher, 2020).

Cyclic sums of comparative indices and their applications

“…Further applications of the comparative index can be found in the spectral theory of (1.1) and (1.2), see [9, Chapters 5,6], [13] and the reference given therein. In the recent publication [25] the notion of the comparative index was connected with the traditional Lidskii angles [22] for symplectic matrices, in [15] we use the comparative index defining the so-called oscillation numbers (see [13,14]) for continuous Lagrangian paths and connect the oscillation numbers with the Maslov index in [18].…”