Non-radial scattering theory for nonlinear Schrödinger equations with potential

Scattering and blowup beyond the mass-energy threshold for the cubic NLS with a potential

“…Remark 1.1. As pointed out in Dinh [5,Remark 1.1], there exists a typical example of a potential satisfying (1.2) and (1.3), called the Yukawatype potential V (x) = c|x| −σ e −a|x| , c ∈ R, σ ∈ (0, 2), a > 0.…”

“…x (K × R 3 ) ∩ L 5 t,x (K × R 3 ) for every compact interval K ⊂ I and obeys the Duhamel formula (1.5) u(t) = e −itH u 0 − i t 0 e −i(t−s)H (|u| 2 u)(s) ds for each t ∈ I. We call I the lifespan of u.…”

Mass-energy threshold dynamics for the focusing NLS with a repulsive inverse-power potential

“…For more details, we refer to Section 6. [4,Theorem 7.2. ], using the argument of Dodson and Murphy [5] the author shows that under condition (1.4)…”