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].
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].
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.
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) Wβ(x)=N-1+2βW(Nβx) , for any β>0 β>0 , or to be given by VN(x)=e2NV(eNx) VN(x)=e2NV(eNx) , for some spherical symmetric, nonnegative and compactly supported 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 VN VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.
We consider the dynamics of N interacting bosons initially exhibiting Bose–Einstein condensation. Due to an external trapping potential, the bosons are strongly confined in two spatial directions, with the transverse extension of the trap being of order ε. The non-negative interaction potential is scaled such that its scattering length is positive and of order (N/ε2)−1 and the range of the interaction scales as (N/ε2)−β for β ∈ (0, 1). We prove that in the simultaneous limit N → ∞ and ε → 0, the condensation is preserved by the dynamics and the time evolution is asymptotically described by a cubic defocusing nonlinear Schrödinger equation in one dimension, where the strength of the nonlinearity depends on the interaction and on the confining potential. This is the first derivation of a lower-dimensional effective evolution equation for singular potentials scaling with β≥12 and lays the foundations for the derivation of the physically relevant one-dimensional Gross–Pitaevskii equation (β = 1). For our analysis, we adapt an approach by Pickl to the problem with strong confinement.
This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory. Mean field equations provide an approximate ansatz for the description of interacting many-particle systems. In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Next-order corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a many-body Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its next-order corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters. In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the N-particle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space. Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding l particles out of their total number N not in the condensate is exponentially small in the number l. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties. The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas. In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.
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 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(1/N).