Generalized Maslov Indices for Non-Hamiltonian Systems

Renormalized oscillation theory for regular linear non-Hamiltonian systems

“…Finally, we will say that a map g : Remark 1.1. The terminology "invariant" is taken from [2], where it arises naturally as the condition that a path in P ( n (R 2n )) (i.e., the projective space of all one-dimensional subspaces of the wedge space n (R 2n )) associated to the flow t → g (t) lies entirely in the Maslov-Arnold space introduced in [2]. While this notion of the Maslov-Arnold space is critical to the development of [2], we will only use it indirectly here, and so will omit a precise definition.…”

“…In the event that the flow t → g (t) is invariant on [a, b] with respect to ω 1 and ω 2 , the generalized Maslov index of [2] can be computed as the winding number in projective space RP 1 of the map t → [ω 1 (g 1 (t), . .…”

“…, g n (t))] (1.9) through [0 : 1] (with appropriate conventions taken for counting arrivals and departures; see Section 2 below). Following the convention of [2], we denote the generalized Maslov index as Ind(• • • ), though our specific notation is adapted from [17,18], leading to Ind(g (•), q; [a, b]); i.e., Ind(g (•), q; [a, b]) is a directed count of the number of times the subspace g (t) has nontrivial intersection with q, counted without multiplicity, as t increases from a to b.…”

“…Given ω 1 as specified in (1.14), and ω 2 such that ker ω 2 = ∆ (not necessarily as in (1.16)), we will be particularly interested in computing the generalized Maslov index along the boundary of [0, 1] × [λ 1 , λ 2 ] (see Figure 4.1, below, in which we follow a long-standing convention of taking the axis associated with the spectral parameter to be horizontal). Following the notation of [2], we will denote this quantity m, and precisely it follows from path additivity of the generalized Maslov index (as discussed in Section 2) that…”

“…Plan of the paper. In Section 2, we discuss the generalized Maslov index of [2], with an emphasis on properties that will be necessary for our analysis, and in Section 3 we discuss the application of renormalized oscillation theory in the current setting. In Section 4 we prove Theorem 1.1, and in Section 5 we develop a framework for checking the invariance assumption of Theorem 1.1 and computing the value m in particular cases.…”