Perturbation of the nonlinear Schrodinger equation from a linear perspective: Vector-valued singular integrals from a scalar perspective.

摘要

Chapter 1. In this section we study the asymptotic properties of solutions to the equation f′′x +fx 2fx+Ef x-Vxf x=0, E0, treating the potential V(x) as a perturbation. The linear part of this equation has been analyzed quite extensively, yielding results about the spectral theory of one-dimensional Schrödinger operators. Our main result is the following statement: If |V(x)| ≤ C|x|−β, β > $32$, then, for every energy E > 0, almost every perturbed solution asymptotically approaches a solution to the unperturbed equation.; This differs from comparable results in the linear setting in two respects. First, the lower limit on β is $32$ as opposed to ½ for the linear Schrödinger operator. Second, the asymptotic description holds for almost every solution at every energy rather than holding for every solution at almost every energy.; Both discrepancies arise from the same basic consequence of nonlinearity: even at a fixed energy E, solutions to the unperturbed nonlinear equation may oscillate with different periods. Their peaks and troughs do not remain in tandem as x → ∞ but instead drift in and out of phase with one another. The stronger decay condition on V is necessary to control the rate at which perturbed solutions “drift” in relation to any given unperturbed solution.; The structure of the almost-everywhere conclusion is derived from a Fourier analysis argument in which the period of unperturbed solutions plays a crucial role. In the linear case, all solutions to the unperturbed equation with a fixed energy have the same period, therefore the value of E is the determining factor. In the nonlinear case, where the period of unperturbed solutions depends on additional parameters, the set of exceptional solutions does not possess such a transparent structure.; Chapter 2. The Muckenhoupt A_p condition concisely describes the class of measures dμ = w(x)dx for which several noteworthy operators (the Hilbert Transform and Hardy-Littlewood maximal function, to name two) are bounded on Lp(dμ). In the setting of vector-valued functions, weights w( x) are defined to take values in the space of nonnegative self-adjoint matrices, and it is natural to ask what condition on matrix weights will continue to guarantee the Lp-boundedness of these operators.; The theory of matrix A_p weights, as developed by Nazarov, Treil, and Volberg, answers this question for singular integral operators. Their approach is quite different from the classical scalar method, eschewing the Hardy-Littlewood maximal function and distributional estimates in favor of Carleson measures and a dyadic embedding operator. Some of the results thus obtained were novel even when restricted to scalars as a special case. In the current work we attempt to perform the opposite feat: employing techniques commonly associated with the scalar case to gain new perspectives on matrix A_p