Transport in the One-dimensional Schrödinger Equation

We prove a dispersive estimate for the Schrödinger equation on the real line, mapping between weighted Lp spaces with stronger time-decay (|t|− 3 2 versus |t|− 1 2 ) than is possible on unweighted spaces. To satisfy this bound, the long-term behavior of solutions must include transport away from the origin. Our primary requirements are that 〈x〉3V be integrable and −∆+V not have a resonance at zero energy. If a resonance is present (for example, in the free case), similar estimates are valid after projecting away from a rank-one subspace corresponding to the resonance. In one dimension, the linear propagator of the free Schrödinger equation is given by the explicit convolution e−it∆ψ(x) = 1 √ −4πi t ∫ R e−i |x−y|2 4t ψ(y) dy. This gives rise immediately to the dispersive estimate (1) ‖eψ‖∞ ≤ (4π|t|)− 1 2 ‖ψ‖1. Such an estimate cannot be true in general for the perturbed operator H = −∆ + V (x). Even small perturbations of the Laplacian may lead to the formation of bound states, i.e. functions fj ∈ L satisfying Hfj = −Ejfj . Bound states with strictly negative energy are known to possess exponential decay, hence they belong to the entire range of L(R), 1 ≤ p ≤ ∞. For each of these bound states fj , the associated evolution efj = e jfj clearly violates (1). It is well known [3, 10] that if V ∈ L(R) then the pure-point spectrum of H consists of at most countably many eigenvalues−Ej < 0. The absolutely continuous spectrum of H is the entire positive half-line, and there is no singular continuous spectrum. Bound states can therefore be removed easily via a spectral projection, suggesting that one should look instead for dispersive estimates of the form (2) ‖ePac(H)ψ‖∞ . |t|− 1 2 ‖ψ‖1. The condition V ∈ L does not always guarantee regularity at the endpoint of the continuous spectrum. We say that zero is a resonance of H if there exists a bounded solution to the equation Hf = 0. Since resonances are not removed by the spectral projection Pac(H), the validity of dispersive estimates invariably depends on whether zero is a resonance ofH. Weder [12] and Goldberg-Schlag [5] have shown that (2) holds for all potentials with (1 + |x|)V ∈ L, and that (1 + |x|)V ∈ L suffices provided zero is not a resonance. 2000 Mathematics Subject Classification. Primary: 35Q40; Secondary: 34L25.