Time decay for solutions of Schr�dinger equations with rough and time-dependent potentials

Abstract

|V (y)| |x − y| dy < 4π.We also establish the dispersive estimate with an ε-loss for large energies provided V K + V 2 < ∞. Finally, we prove Strichartz estimates for the Schrödinger equations with potentials that decay like |x| −2−ε in dimensions n ≥ 3, thus solving an open problem posed by Journé, Soffer, and Sogge.

Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions

“…The Strichartz estimates stated in Theorem 1.1 will be proved using Proposition 2.1 below, which was proved in [18], see Theorem 4.1 in that paper. It is based on Kato's notion of smoothing operators, see [13].…”

“…Instead, we adopt an argument introduced in [18], where the validity of Strichartz inequalities is instead derived from Kato's theory of smooth perturbations. This paper is related to our three-dimensional paper [6], where a result similar to Theorem 1.1 was proved but under much stronger conditions on A, V , both in terms of decay as well as regularity.…”

Decay estimates for the wave and Dirac equations with a magnetic potential

“…Several results are available for the equations i∂ t u − ∆u + V (x)u = 0, u + V (x)u = 0. We cite among the others [8], [15], [16], [19], [32] and the recent survey [33] for Schrö-dinger; and [5], [6], [10], [12], [13] for the wave equation. We must also mention the wave operator approach of Yajima (see [2], [39], [40], [41]) which permits to deal with the above equations in a unified way, although under nonoptimal assumptions on the potential in dimensions 1 and 3.…”

Solvability Conditions for a Linearized Cahn-Hilliard Equation of Sixth Order

“…By means of Lemma 2.3 of [18], under our Assumption 1 above on the potential function, the operator −∆ − V (x) − a is self-adjoint and unitarily equivalent to −∆ − a on L 2 (R 3 ) via the wave operators (see [10], [14]) Ω ± := s − lim t→∓∞ e it(−∆−V ) e it∆ with the limit understood in the strong L 2 sense (see e.g. [13] p.34, [4] p.90).…”