Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation
We consider the dynamics of N bosons in 1D. We assume that the pair interaction is attractive and given by N β - 1 V ( N β ) . where ∫ V ⩽ 0 . We develop new techniques in treating the N -body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new...
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| Published in | Archive for rational mechanics and analysis Vol. 221; no. 2; pp. 631 - 676 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.08.2016
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0003-9527 1432-0673 |
| DOI | 10.1007/s00205-016-0970-6 |
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| Abstract | We consider the dynamics of
N
bosons in 1D. We assume that the pair interaction is attractive and given by
N
β
-
1
V
(
N
β
)
.
where
∫
V
⩽
0
. We develop new techniques in treating the
N
-body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the 1D focusing cubic NLS with a quadratic trap as the
N
→
∞
limit of the
N
-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system. |
|---|---|
| AbstractList | We consider the dynamics of
N
bosons in 1D. We assume that the pair interaction is attractive and given by
N
β
-
1
V
(
N
β
)
.
where
∫
V
⩽
0
. We develop new techniques in treating the
N
-body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the 1D focusing cubic NLS with a quadratic trap as the
N
→
∞
limit of the
N
-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We consider the dynamics of N bosons in 1D. We assume that the pair interaction is attractive and given by ... where ... We develop new techniques in treating the N-body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the 1D focusing cubic NLS with a quadratic trap as the ... limit of the N-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system. |
| Author | Holmer, Justin Chen, Xuwen |
| Author_xml | – sequence: 1 givenname: Xuwen surname: Chen fullname: Chen, Xuwen email: chenxuwen@math.brown.edu organization: Department of Mathematics, University of Rochester – sequence: 2 givenname: Justin surname: Holmer fullname: Holmer, Justin organization: Department of Mathematics, Brown University |
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| Snippet | We consider the dynamics of
N
bosons in 1D. We assume that the pair interaction is attractive and given by
N
β
-
1
V
(
N
β
)
.
where
∫
V
⩽
0
. We develop new... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We consider the dynamics of N bosons in 1D. We assume that the pair interaction is... |
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| SubjectTerms | Classical Mechanics Complex Systems Derivation Derivatives Dynamical systems Dynamics Estimates Fluid- and Aerodynamics Mathematical and Computational Physics Nonlinear dynamics Physics Physics and Astronomy Schroedinger equation Texts Theoretical |
| Title | Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation |
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