Decay estimates for Schrödinger operators

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.…”

Inverse scattering for the non‐linear Schrödinger equation: Reconstruction of the potential and the non‐linearity

“…This result and the intertwining relations for the wave operators, e −itH P c = W ± e −itH0 W * ± , imply that the following L p − Lṕ estimate follows from the corresponding result for H 0 (see [24]): [8].…”

Tosio Kato’s work on non-relativistic quantum mechanics: part 2

“…The first estimates of the type (16.27) were found by Schonbek [542] in 1979 who considered ν = 3, p = 1 and V small. The first general result for ν ≥ 3 and V so that H has neither an eigenvalue nor resonance at zero energy were in a classic paper of Journé et al [294] (see also Schonbek-Zhou [543]). …”