We consider the 1-dimensional cubic nonlinear Schrödinger equation with a large external
potential
with no
bound states. We prove global regularity and quantitative bounds for small solutions under mild
assumptions on
.
In particular, we do not require any differentiability of
and
make spatial decay assumptions that are weaker than those found in the literature
(see for example work of Delort (2016), Naumkin (2016) and Germain et al. (2018)).
We treat both the case of generic and nongeneric potentials, with some additional
symmetry assumptions in the latter case.
Our approach is based on the combination of three main ingredients: the
Fourier transform adapted to the Schrödinger operator, basic bounds on
pseudodifferential operators that exploit the structure of the Jost function, and
improved local decay and smoothing-type estimates. An interesting aspect of
the proof is an “approximate commutation” identity for a suitable notion
of a vector field, which allows us to simplify the previous approaches and
extend the known results to a larger class of potentials. Finally, under our
weak assumptions we can include the interesting physical case of a barrier
potential as well as recover the result of Masaki et al. (2019) for a delta
potential.
PDF Access Denied
We have not been able to recognize your IP address
18.221.174.248
as that of a subscriber to this journal.
Online access to the content of recent issues is by
subscription, or purchase of single articles.
Please contact your institution's librarian suggesting a subscription, for example by using our
journal-recommendation form.
Or, visit our
subscription page
for instructions on purchasing a subscription.
