Long Time Existence for Magnetic Nonlinear Schrödinger Equations

Shijun Zheng

Long Time Existence for Magnetic Nonlinear Schrödinger Equations

Mathematical Sciences

Research output: Contribution to conference › Presentation

Abstract

<div class="line" id="line-5"> 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&dot; = ξ ξ&dot;=-∇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).</div>

Original language	American English
State	Published - Mar 7 2015
Event	Spring Eastern Sectional Meeting of the American Mathematical Society (AMS) - Duration: Mar 19 2016 → …

Conference

Conference	Spring Eastern Sectional Meeting of the American Mathematical Society (AMS)
Period	03/19/16 → …

Keywords

Blowup
Electromagnetic potentials
Energy density
Global existence
Lorentz force
NLS mechanism
Newton's Law
Schrodinger operator
Semiclassical regime
Uler-Lagrange equation

DC Disciplines

Mathematics

Cite this

@conference{a6f9e53223c24154aa1e9ef11065db93,

title = "Long Time Existence for Magnetic Nonlinear Schr{\"o}dinger Equations",

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&dot; = ξ ξ&dot;=-∇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).",

keywords = "Blowup, Electromagnetic potentials, Energy density, Global existence, Lorentz force, NLS mechanism, Newton's Law, Schrodinger operator, Semiclassical regime, Uler-Lagrange equation",

author = "Shijun Zheng",

year = "2015",

month = mar,

day = "7",

language = "American English",

note = "Spring Eastern Sectional Meeting of the American Mathematical Society (AMS) ; Conference date: 19-03-2016",

}

TY - CONF

T1 - Long Time Existence for Magnetic Nonlinear Schrödinger Equations

AU - Zheng, Shijun

PY - 2015/3/7

Y1 - 2015/3/7

N2 - 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&dot; = ξ ξ&dot;=-∇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).

AB - 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&dot; = ξ ξ&dot;=-∇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).

KW - Blowup

KW - Electromagnetic potentials

KW - Energy density

KW - Global existence

KW - Lorentz force

KW - NLS mechanism

KW - Newton's Law

KW - Schrodinger operator

KW - Semiclassical regime

KW - Uler-Lagrange equation

UR - http://www.ams.org/amsmtgs/2225_abstracts/1107-35-319.pdf

M3 - Presentation

T2 - Spring Eastern Sectional Meeting of the American Mathematical Society (AMS)

Y2 - 19 March 2016

ER -

Long Time Existence for Magnetic Nonlinear Schrödinger Equations

Abstract

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