On the characterization of the smoothness of skew-adjoint potentials in periodic Dirac operators

Abstract

In this paper we consider periodic Dirac operators with skew-adjoint potentials in a large class of weighted Sobolev spaces. We characterize the smoothness of such potentials by asymptotic properties of the periodic spectrum of the corresponding Dirac operators.

Birkhoff Coordinates for the Focusing NLS Equation

“…The main result of this paper is the following N by the weighted sequence space l 2 ω -see [9]. Theorem 1.1 can be used to obtain a KAM-result for the focusing NLS equation of the type obtained in [5] for the defocusing NLS equation.…”

New Estimates of the Nonlinear Fourier Transform for the Defocusing NLS Equation

“…We point out that these coefficients depend on λ and ϕ. It has been observed in [6,16] that these coefficients reflect certain symmetries of the Fourier coefficients of ϕ. We only need the fact that a + n and a − n coincide.…”

“…The decay properties of the actions I n are known to be closely related to the regularity properties of ϕ -c.f. [14,6,16]. We need to quantify this relationship by providing two sided estimates of the Sobolev norms of ϕ in terms of weighted ℓ 1 -norms of I(ϕ).…”

Characterization of the potential smoothness of one-dimensional Dirac operator subject to general boundary conditions and its Riesz basis property

“…Similarly, the case (ii) in (40) also gives the existence of a positive lower bound for | f n ,g n |. Now, we consider the equation (18). First, we note that t n = −τ n .ℓ 0 (f n ) and |ℓ 0 (f n )| ≥ |ℓ 0 (f 0 n )|−κ n ≥ C 0 −κ n , where C 0 is a positive number depending on general boundary conditions and κ n ≤ C 0 /2.…”

“…We also have (17) LG n ,g n = L bc G n ,g n = G n , (L bc ) * g n = G n , µ ngn = µ n G n ,g n . The equality of (16) and (17) leads to (18) (µ n − λ + n ) G n ,g n = t n (ξ n f n ,g n − γ n ϕ n ,g n ). The equation (18) is important that our proof of the estimation for |µ n − λ + n | will be based on the approximations for each remaining term in (18).…”