A construction of the fundamental solution for the Schrödinger equation

Semiclassical Asymptotics for Weakly Nonlinear Bloch Waves

“…That article is for the case W ≡ 0; the proof uses Strichartz inequalities, following from dispersion estimates. When W is smooth, real-valued and sub-quadratic, the same dispersion estimates are available ( [17,18]), and they imply the same Strichartz inequalities ( [28]). Another difference with [25] is that the Galilean operator x + it∇ x commutes with i∂ t + 1 2 ∆, but in general not with i∂ t + 1 2 ∆ − W .…”

Nonlinear Schrödinger equation with time dependent potential

“…To solve (1.2), even locally in time, one needs to work in Σ, and not only H 1 ; symmetry is needed on physical and frequency sides, unless V is sublinear [9]. Local existence in Σ then follows from the dispersive estimates in [20,21]; one can work as in the case V ≡ 0 (where it is possible to work in H 1 (R d ) only). Instead of considering only (u,∇u) as the unknown function, one can consider (u,∇u,xu).…”

“…Since only bounded time intervals are considered in [20,21], we give a more precise treatment of this result in §2 in order to consider global in time solutions. We have been careful in the statement of Proposition 1.6 not to write that T depends only on u 0 Σ .…”

“…In view of [20,21], local in time Strichartz estimates are available. Therefore, one can reproduce the original proof of [37] (see also [13,34]), in order to infer the result.…”

Representation formula for stochastic Schrödinger evolution equations and applications

“…Such oscillatory integral operators have been studied by many authors in the context of deterministic Schrödinger equations (see e.g. [13,14,15,18,20,28]). In the present paper we follow Yajima [28] who derived dispersive estimates for Schrödinger equations with magnetic elds.…”