A rigorous derivation of the Hamiltonian structure for the nonlinear Schrodinger equation

被引:9
作者
Mendelson, Dana [1 ]
Nahmod, Andrea R. [2 ]
Pavlovic, Natasa [3 ]
Rosenzweig, Matthew [3 ]
Staffilani, Gigliola [4 ]
机构
[1] Univ Chicago, Dept Math, 5734 S Univ Ave, Chicago, IL 60637 USA
[2] Univ Massachusetts, Dept Math, 710 N Pleasant St, Amherst, MA 01003 USA
[3] Univ Texas Austin, Dept Math, 2515 Speedway,Stop C1200, Austin, TX 78712 USA
[4] MIT, Dept Math, 77 Massachusetts Ave, Cambridge, MA 02139 USA
关键词
Nonlinear Schrodinger; Infinite-dimensional Hamiltonian system; Quantum many-body systems; GROSS-PITAEVSKII HIERARCHY; CLASSICAL LIMIT; FIELD LIMIT; QUANTUM; DYNAMICS; UNIQUENESS;
D O I
10.1016/j.aim.2020.107054
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We consider the cubic nonlinear Schrodinger equation (NLS) in any spatial dimension, which is a well-known example of an infinite-dimensional Hamiltonian system. Inspired by the knowledge that the NLS is an effective equation for a system of interacting bosons as the particle number tends to infinity, we provide a derivation of the Hamiltonian structure, which is comprised of both a Hamiltonian functional and a weak symplectic structure, for the nonlinear Schrodinger equation from quantum many-body systems. Our geometric constructions are based on a quantized version of the Poisson structure introduced by Marsden, Morrison and Weinstein [24] for a system describing the evolution of finitely many indistinguishable classical particles. (C) 2020 Elsevier Inc. All rights reserved.
引用
收藏
页数:115
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