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arXiv:2308.09674 · Journal of Functional Analysis

Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction

Riccardo Adami, Jinyeop Lee

Published Feb 10, 2025• 3 citations
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Abstract

We derive an effective equation for the dynamics of many identical bosons in dimension one in the presence of a tiny impurity. The interaction between every pair of bosons is mediated by the impurity through a positive three-body potential. Assuming a simultaneous mean-field and short-range scaling with the short-range proceeding slower than the mean-field, and choosing an initial fully condensed state, we prove propagation of chaos and obtain an effective one-particle Schrödinger equation with a defocusing nonlinearity concentrated at a point. More precisely, we prove convergence of one-particle density operators in the trace-class topology and estimate the fluctuations as superexponential. This is the first derivation of the so-called nonlinear delta model, widely investigated in the last decades, as a phenomenological model for several physical phenomena .

DOI
10.1016/j.jfa.2025.110866

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We derive an effective equation for the dynamics of many identical bosons in dimension one in the presence of a tiny impurity. The interaction between every pair of bosons is mediated by the impurity through a positive three-body potential. Assuming a simultaneous mean-field and short-range scaling with the short-range proceeding slower than the mean-field, and choosing an initial fully condensed state, we prove propagation of chaos and obtain an effective one-particle Schrödinger equation with a defocusing nonlinearity concentrated at a point. More precisely, we prove convergence of one-particle density operators in the trace-class topology and estimate the fluctuations as superexponential. This is the first derivation of the so-called nonlinear delta model, widely investigated in the last decades, as a phenomenological model for several physical phenomena .
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.
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by a3β−1V(aβ⋅)a^{3β-1}V(a^β\cdot)a3β−1V(aβ⋅) where ∫V⩽0\int V\leqslant 0∫V⩽0 and aaa matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D NNN-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to $\infty $. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.
We consider the dynamics of N interacting Bosons in three dimensions which are strongly confined in one or two directions. We analyze the mean field scaling and the nonlinear Schrödinger scaling of the interaction potential w and choose the initial wavefunction to be close to a product wavefunction. For both scalings we prove that in the mean field limit the dynamics of the N-particle system is described by a nonlinear equation in two or one dimensions. In the case of the mean field scaling this equation is the Hartree equation and for the second scaling the nonlinear Schrödinger equation. In both cases we obtain explicit bounds for the rate of convergence of the N-particle dynamics to the one-particle dynamics.
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.
We study the mean field limit of nonrelativistic quantum many-boson systems with delta potential in one-dimensional space. Such problem is related to the semiclassical limit of a second quantized Hamiltonian with an interaction given by a quartic Wick product. In this framework, we show that the evolution of coherent states is semiclassically given by squeezed coherent states under the action of a time-dependent affine Bogoliubov transformation. Results similar to those stated by Hepp [Comm. Math. Phys. 35 (1974), 265–277] and Ginibre-Velo [Comm. Math. Phys. 66 (1979), 37–76 and 68 (1979), 45–68] are proved. Furthermore, we show propagation of chaos for Schrödinger dynamics in the mean field limit using the argument of Rodnianski–Schlein [Comm. Math. Phys. 291 (2009), 31–61]. Thus, we provide a derivation of the cubic NLS equation in one dimension.
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 lll particles out of their total number N not in the condensate is exponentially small in the number lll. 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.