Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems

By L. Erdős, B. Schlein, Horng-Tzer Yau. 2005

336 citations recorded by the source corpus.

We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schrödinger equation in a suitable scaling limit. The result is extended to k-particle density matrices for all positive integer k.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/87d54844ff3cdef5d53e00f70c1a5e9eea6ad7bc

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In Artificial Intelligence

In Atomic and Molecular Physics, and Optics

Citations in the corpus

Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I. (2009, methodology): “This approach has been highly successful, even for very singular potentials, in the work of Elgart, Erdős, Schlein and Yau [6, 7, 8, 9, 10, 11].”
A review of the mean field limits for Vlasov equations (2014, background): “• Quantum mechanics and the derivation of non linear Schrödinger equations (in particular Schrödinger-Poisson and Gross-Pitaevskii) from linear, N -particles Schrödinger; see for example [9], [8], [61], [62] and the references therein.”
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics (2007, methodology): “In [3, 5, 6], models described by the Hamiltonian HN = N ∑”
Quantitative Derivation of the Gross‐Pitaevskii Equation (2012, methodology): “This is in contrast with the results of [7, 8, 9, 10], which can also be applied to completely factorized initial data.”
Kac’s program in kinetic theory (2011, methodology): “The BBGKY hierarchy method revisited The so-called BBGKY hierarchy method (Bogoliubov, Born, Green, Kirkwood and Yvon) is very popular in physics and mathematics for studying many-particle systems: see for instance [3] where this approach is used for Kac’s master equation for hard spheres, or see, a”
Derivation of the Gross‐Pitaevskii hierarchy for the dynamics of Bose‐Einstein condensate (2004, background): “In [5], we also prove that the GP hierarchy has a unique solution in an appropriate Sobolev space.”
Mean–Field Evolution of Fermionic Systems (2013, methodology): “More recently, these techniques have been applied in [10, 11, 12, 13] to systems of bosons in the so called Gross-Pitaevskii limit; in this case, the interaction is scaled so that its range and its scattering length are both of the order 1/N .”
A Simple Derivation of Mean Field Limits for Quantum Systems (2009, background): “Derivations of such mean field equations from the microscopic N body Schrödinger evolution are usually done for the reduced one particle density and are naturally based on hierarchies [16, 3, 2, 4, 5, 1].”