Dichotomy of Systems and Invertibility of Linear Ordinary Differential Operators

Relative Morse indices, Fredholm indices, and group velocities

“…Palmer proved in [9,10] that T is Fredholm on C 0 unif if and only if the ordinary differential equation T U = 0 has exponential dichotomies on R + and on R − . Ben-Artzi and Gohberg [1] proved the same result for L 2 spaces. Alternatively, the equivalence of exponential dichotomies on R ± and Fredholm properties on L 2 was proved in [12] for a far more general class of operators that may depend on additional independent variables provided c is a constant; if there are no additional variables present as in our setting, the proof in [12] works for the operators considered here 1 .…”

“…have Morse index i = ∞). This case is of interest when formulating travelling-wave problems u ξξ + cu ξ + ∆ y u + f (u) = 0, (ξ, y) ∈ R × Ω on cylinders as dynamical systems in ξ or when studying time-periodic travellingwave solutions u = u(x − ct, ωt), u(ξ, τ ) = u(ξ, τ + 2π) of the reaction-diffusion system (1).…”

Evans Functions, Jost Functions, and Fredholm Determinants

“…Indeed, by a well-known Dichotomy Theorem (sometimes called Palmer's theorem), the operator G is Fredholm on L 2 (R) d if and only if (1.1) has exponential dichotomies Q − on R − and Q + on R + ; see [3], [48], [49], [56] or [57, Theorem 3.2], and also [41], [53] for more recent versions of the dichotomy theorem. ✸ Next, we will discuss the Bohl exponents and exponential splittings for the perturbed equation (1.2).…”

Functionals on function and sequence spaces connected with the exponential stability of evolutionary processes

“…This idea was later extensively developed for the discrete-time systems in the infinite-dimensional case by Ch. V. Coffman and J.J. Schäffer in 1967 [2] and D. Henry in 1981 [4] and more recently we refer the readers to the papers due to A. Ben-Artzi [1], I. Gohberg [1], M. Pinto [12], J. P. La Salle [5].…”