#### Title

Long Time Existence for Magnetic Nonlinear Schrödinger Equations

#### Document Type

Presentation

#### Publication Date

3-7-2015

#### Abstract

Denote by L=-½∇2A + V the Schrödinger operator with electromagnetic potentials, where A is sublinear and V subquadratic. The NLS mechanism generated by L in the semiclassical regime obeys the Newton's law x˙ = ξ ξ˙=-∇V(x)-ξ X B(x) in the transition from quantum to classical mechanics, which can be derived by the Euler-Lagrange equation. Here B=∇ X A is the magnetic field induced by A and the Lorentz force is given by -ξ X B. the energy density H (t := ½ |ξ(t)|2 + V (x(t)) is conserved in time. We study the fundamental solution for e-itL and consider the threshold for the global existence and blowup for the NLS. (Received January 18, 2015).

#### Sponsorship/Conference/Institution

Spring Eastern Sectional Meeting of the American Mathematical Society (AMS)

#### Location

Washington, DC

#### Recommended Citation

Zheng, Shijun.
2015.
"Long Time Existence for Magnetic Nonlinear Schrödinger Equations."
*Mathematical Sciences Faculty Presentations*.
Presentation 550.

https://digitalcommons.georgiasouthern.edu/math-sci-facpres/550