Long Time Existence for Magnetic Nonlinear Schrödinger Equations
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).
Spring Eastern Sectional Meeting of the American Mathematical Society (AMS)
"Long Time Existence for Magnetic Nonlinear Schrödinger Equations."
Mathematical Sciences Faculty Presentations.