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Methods of Harmonic Analysis Applied to Bose-Einstein Condensation

Xuwen Chen

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

This dissertation studies questions whose origin is in quantum statistical mechanics and is concerned about the evolution of large numbers of quantum spinless, interacting particles. More specifically, we study the analysis of the $N$-particle linear Schr\"{o}dinger equation as $N\rightarrow \infty $ and discuss rigorously how the $1$-body nonlinear Schr\"{o}dinger equation comes from this limit process. Such problems arise in Bose-Einstein condensation. In the first part of this dissertation, we consider the 2d and 3d many body Schr\"{o}dinger equations in the presence of anisotropic switchable quadratic traps. We extend and improve the collapsing estimates in Klainerman-Machedon [29] and Kirkpatrick-Schlein-Staffilani [27]. Combining with an anisotropic version of the generalized lens transform as in Carles [3], we offer a rigorous derivation of the cubic NLS with anisotropic switchable quadratic traps in 2d through an appropriately modified procedure in Elgart-Erd\"{o}s-Schlein-Yau [12, 13, 14, 15, 16, 17, 18] which is based on a kinetic hierarchy. For the 3d case, we establish the uniqueness of the corresponding Gross-Pitaevskii hierarchy without the assumption of factorized initial data. In the second part of this thesis, we consider the Hamiltonian evolution of $N$ weakly interacting Bosons. Assuming triple collisions, its mean field approximation is given by a quintic Hartree equation. We construct a second-order correction to the mean field approximation using a kernel $k(t,x,y)\ $ describing pair creation and derive an evolution equation for $k$. We show the global existence for the resulting evolution equation for the correction and establish an apriori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work of Rodnianski and Schlein [35], and Grillakis, Machedon and Margetis [21,22], where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order of $O\left( 1/\sqrt{N}\right) .$

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This dissertation studies questions whose origin is in quantum statistical mechanics and is concerned about the evolution of large numbers of quantum spinless, interacting particles. More specifically, we study the analysis of the NNN-particle linear Schr"{o}dinger equation as N→∞N\rightarrow \infty N→∞ and discuss rigorously how the 111-body nonlinear Schr"{o}dinger equation comes from this limit process. Such problems arise in Bose-Einstein condensation. In the first part of this dissertation, we consider the 2d and 3d many body Schr"{o}dinger equations in the presence of anisotropic switchable quadratic traps. We extend and improve the collapsing estimates in Klainerman-Machedon [29] and Kirkpatrick-Schlein-Staffilani [27]. Combining with an anisotropic version of the generalized lens transform as in Carles [3], we offer a rigorous derivation of the cubic NLS with anisotropic switchable quadratic traps in 2d through an appropriately modified procedure in Elgart-Erd"{o}s-Schlein-Yau [12, 13, 14, 15, 16, 17, 18] which is based on a kinetic hierarchy. For the 3d case, we establish the uniqueness of the corresponding Gross-Pitaevskii hierarchy without the assumption of factorized initial data. In the second part of this thesis, we consider the Hamiltonian evolution of NNN weakly interacting Bosons. Assuming triple collisions, its mean field approximation is given by a quintic Hartree equation. We construct a second-order correction to the mean field approximation using a kernel k(t,x,y) k(t,x,y)\ k(t,x,y)  describing pair creation and derive an evolution equation for kkk. We show the global existence for the resulting evolution equation for the correction and establish an apriori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work of Rodnianski and Schlein [35], and Grillakis, Machedon and Margetis [21,22], where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order of O(1/N).O\left( 1/\sqrt{N}\right) .O(1/N​).
This thesis proves certain results concerning an important question in non-equilibrium quantum statistical mechanics which is the derivation of effective evolution equations approximating the dynamics of a system of large number of bosons initially at equilibrium (ground state at very low temperatures). The dynamics of such systems are governed by the time-dependent linear many-body Schroedinger equation from which it is typically difficult to extract useful information due to the number of particles being large. We will study quantitatively (i.e. with explicit bounds on the error) how a suitable one particle non-linear Schroedinger equation arises in the mean field limit as number of particles N → ∞ and how the appropriate corrections to the mean field will provide better approximations of the exact dynamics. In the first part of this thesis we consider the evolution of N bosons, where N is large, with two-body interactions of the form N³ᵝv(Nᵝ⋅), 0≤β≤1. The parameter β measures the strength and the range of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [18,19] by Grillakis-Machedon-Margetis. We extend the results for 0 ≤ β < 1/3 in [19, 20] to the case of β < 1/2 and obtain an error bound of the form p(t)/Nᵅ, where α>0 and p(t) is a polynomial, which implies a specific rate of convergence as N → ∞. In the second part, utilizing estimates of the type discussed in the first part, we compare the exact evolution with the mean field approximation in the sense of marginals. We prove that the exact evolution is close to the approximate in trace norm for times of the order o(1)√N compared to log(o(1)N) as obtained in Chen-Lee-Schlein [6] for the Hartree evolution. Estimates of similar type are obtained for stronger interactions as well.
This thesis investigates the rigorous derivation of non-linear effective equations describing the time evolution of Bose-Einstein condensation from the N-particle linear Schrödinger equation, focusing on the one-particle reduced density matrix. The first chapter outlines the mathematical framework of many-body quantum mechanics, including the Fock space formalism. The second chapter studies the mean-field regime, proving that the condensed state converges to the non-linear Hartree equation by analyzing fluctuation dynamics to quantify the convergence rate. The third chapter addresses the physically realistic but mathematically challenging Gross-Pitaevskii regime, characterized by rare but strong interactions. Here, Bogoliubov transformations are used to capture crucial microscopic short-scale correlations, allowing for a quantitative derivation of the cubic non-linear Schrödinger equation. While the preceding chapters focus exclusively on repulsive interactions, the fourth chapter explores the focusing case with attractive potentials, which can lead to finite-time blow-up. Operating in an intermediate scaling regime, the effective focusing cubic non-linear Schrödinger equation is rigorously derived. Since the negative energy cannot bound many-body fluctuations, convergence is proven by approximating the dynamics via a quadratic Hamiltonian, in the spirit of Bogoliubov theory, and avoiding Grönwall estimates through the analysis of a coupled system of one-body linear equations for the one-particle density matrices.
The goal of this thesis is to provide a mathematical rigorous derivation of the Schrödinger-Klein-Gordon equations, the Maxwell-Schrödinger equations and the defocusing cubic nonlinear Schrödinger equation in two dimensions. We study the time evolution of the Nelson model (with ultraviolet cutoff) in a limit where the number N of charged particles gets large while the coupling of each particle to the radiation field is of order N^{−1/2}. At time zero it is assumed that almost all charges are in the same one-body state (a Bose-Einstein condensate) and that the radiation field is close to a coherent state. We show the persistence of condensation over time and prove that the time evolution is approximately described by the Schrödinger-Klein-Gordon system of equations in the large N limit. Subsequently, we consider the spinless Pauli-Fierz Hamiltonian which models the interaction between charged bosons and the quantized electromagnetic field. We discuss the limit previously described and prove that the time evolution is approximated by the Maxwell-Schrödinger equations. To our knowledge, this is the first rigorous result concerning a mean-field limit of the Pauli-Fierz Hamiltonian. We then turn to the evolution of Bose-Einstein condensates in two dimensions and consider N bosons which interact by a repulsive two-body potential. The interaction is given either by N^{−1+2β}V(N^{β}x) with β∈R^{+}_{0} or by e^{2N}V(e^{N}x), for some spherical symmetric, positive and compactly supported V∈L_{∞}(R^{2},R). We prove that the dynamics is approximated by the defocusing two-dimensional cubic nonlinear Schrödinger equation in the large N limit. In case of the exponential scaling, we show that a short-scale correlation structure affects the dynamics of the condensate. This is the first rigorous derivation that considers an exponential scaling of the interaction. All derivations rely on a method developed by Pickl in [Lett. Math. Phys. 97(2), 151–164 (2011)]. The first two results are obtained by an extension of the method to systems which interact with quantized radiation fields. The latter is derived by an appropriate adaption of the proof in three space dimensions [Rev. Math. Phys., 27, 1550005 (2015)]. The crucial insight to derive the Maxwell-Schrödinger equations is to restrict the class of many-body wave functions to a subspace of states whose energy per particle only fluctuates little around the energy functional of the Maxwell-Schrödinger system. To derive the two-dimensional Gross-Pitaevskii equation it is essential to define a measure of condensation which properly incorporates the correlations that arise from the exponential scaling of the interaction. This thesis is based on the preprints [54, 47].
Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N\mathit{N}N. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one-particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error that vanishes as N−1/2\mathit{N}^{-1/2}N−1/2 in the limit of large N\mathit{N}N.
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.