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Matematičeskie struktury i modelirovanie

Derivation of Gross-Pitaevskii Version of Nonlinear Schroedinger Equation from Scale Invariance

Olga M. Kosheleva, Владик Крейнович

Published Jan 1, 2017• 0 citations
Publisher

Abstract

It is known that in the usual 3-D space, the Schroedinger equation can be derived from scale-invariance. In view of the fact that, according to modern physics, the actual dimension of proper space may be different from 3, it is desirable to analyze what happens in other spatial dimensions $D$. It turns out that while for $D\ge 3$ we still get only the Schroedinger's equation, for $D=2$, we also get the Gross-Pitaevskii version of a nonlinear Schroedinger equation that describes a quantum system of identical bosons, and for $D=1$, we also get a new nonlinear version of the Schroedinger equation.

DOI
10.24147/2222-8772.2017.3.35-42

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It is known that in the usual 3-D space, the Schroedinger equation can be derived from scale-invariance. In view of the fact that, according to modern physics, the actual dimension of proper space may be different from 3, it is desirable to analyze what happens in other spatial dimensions DDD. It turns out that while for D≥3D\ge 3D≥3 we still get only the Schroedinger's equation, for D=2D=2D=2, we also get the Gross-Pitaevskii version of a nonlinear Schroedinger equation that describes a quantum system of identical bosons, and for D=1D=1D=1, we also get a new nonlinear version of the Schroedinger equation.
Abstract We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting from an interacting N-particle system of bosons. We consider the interaction potential to be given either by Wβ(x)=N−1+2βW(Nβx)W_\beta (x)=N^{-1+2 \beta }W(N^\beta x)Wβ​(x)=N−1+2βW(Nβx) Wβ(x)=N-1+2βW(Nβx) , for any β>0\beta >0β>0 β>0 , or to be given by VN(x)=e2NV(eNx)V_N(x)=e^{2N} V(e^N x)VN​(x)=e2NV(eNx) VN(x)=e2NV(eNx) , for some spherical symmetric, nonnegative and compactly supported W,V∈L∞(R2,R)W,V \in L^\infty ({\mathbb {R}}^2,{\mathbb {R}})W,V∈L∞(R2,R) W,V∈L∞(R2,R) . In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential VNV_NVN​ VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.
Systems of interest in physics are usually composed by a very large number of interacting particles. At equilibrium, these systems are described by stationary states of the many-body Hamiltonian (at zero temperature, by the ground state). The reaction to perturbations, for example to a change of the external fields, is governed by the time-dependent many-body Schroedinger equation. Since it is typically very difficult to extract useful information from the Schroedinger equation, one of the main goals of non-equilibrium statistical mechanics is the derivation of effective evolution equations which can be used to predict the macroscopic behavior of the system. In these notes, we are going to consider systems of interacting bosons in the so called Gross-Pitaevskii regime, and we are going to show how coherent states and Bogoliubov transformations can be used to approximate the many body dynamics.
We derive rigorously, for both R^2 and [-L, L]^2, the cubic nonlinear Schrodinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.
We derive rigorously, for both R2{\Bbb R}^2R2 and [−L,L]×2[{-}L,L]^{\times 2}[−L,L]×2, the cubic nonlinear Schr\"odinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.
We present effective reduced equations for the study of a binary Bose-Einstein condensate (BEC), where the confining potentials of the two BEC components have distinct asymmetry so that the components belong to different space dimensions as in a recent experiment [G. Lamporesi et al., Phys. Rev. Lett. 104, 153202 (2010)]. Starting from a binary three-dimensional (3D) Gross-Pitaevskii equation (GPE) and using a Lagrangian variational approach we derive a binary effective nonlinear Schroedinger equation with components in different reduced dimensions, e. g., the first component in one dimension and the second in two dimensions as appropriate to represent a cigar-shaped BEC coupled to a disk-shaped BEC. We demonstrate that the effective reduced binary equation, which depend on the geometry of the system, is quite reliable when compared with the binary 3D GPE and can be efficiently used to perform numerical simulation and analytical calculation for the investigation of static and dynamic properties of a binary BEC in mixed dimensions.