arXiv subjects

Jinyeop Lee

Publications and source records attributed to Jinyeop Lee.

At least 19 recordsLinked to original sources

arXiv2026

A Trichotomy for Modified Scattering Across the Yukawa-Coulomb Transition

We study the long-time asymptotics of the three-dimensional Hartree equation with Yukawa potential \[ V_\mu(x)=\frac{e^{-\mu|x|}}{|x|}, \qquad 0\leq\mu\leq1. \] The Coulomb case corresponds to $\mu=0$, while $\mu>0$ introduces the screening length $\mu^{-1}$. In the limit $\mu\to0$ and $t\to\infty$, the asymptotic behavior depends on the comparison between the observation scale $t$ and the screening length $\mu^{-1}$, equivalently on the parameter $\mu t$. This leads to three distinct asymptotic regimes, according as $\mu t\to0$, $\mu t\to L\in(0,\infty)$, or $\mu t\to\infty$, with different modified scattering phases in each case.

math.AP↗

arXiv2026

Modified Scattering for the Time-Dependent Kohn--Sham Equation

We study the long-time behavior of the (critical) Kohn--Sham equation in two and three dimensions, i.e.,\[ \mathrm{i} \partial_t {\gamma} = \Big[-\frac{1}{2}\Delta + \lambda \, |\cdot|^{-1} \ast \rho_{{\gamma}} + \mu \, \rho_{{\gamma}}^{1/d}, {\gamma} \Big] \quad \text{for} \quad d=2,3. \] By introducing a suitable ''square root'' of the density matrix and exploiting the pseudo-conformal transform, we establish global well-posedness for small initial data in an appropriate weighted Schatten norm. We also prove the optimal time decay of the particle density and establish modified scattering for small and localized solutions. In particular, our results provide a resolution to the open problems proposed by Pusateri and Sigal (2021) for the critical and subcritical regimes, rigorously proving their conjectures regarding modified scattering in the critical case and linear scattering in the subcritical cases. Our results place these scattering phenomena in the operator-valued setting of density matrices, thereby extending the classical scalar theory to a broader framework.

math.AP↗

arXiv2026

Commutator Estimates for Low-Temperature Fermi Gases

We investigate the semiclassical regularity of thermal equilibria in the presence of a harmonic potential at low temperature; that is, we obtain the asymptotic behavior of the Schatten norms of commutators of the one-body operators associated with these equilibria and the position and momentum operators. We also obtain upper bounds in the magnetic field case for the Fock-Darwin Hamiltonian. Our estimates, in particular, allow us to observe several regimes depending on the joint behavior of the Planck constant, the temperature, and the strength of the magnetic field.

math-ph↗

arXiv2025

The Semiclassical Limit of the 2D Dirac--Hartree Equation with Periodic Potentials

We study the semiclassical limit of the two-dimensional Dirac--Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac--Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov--Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter.

math-ph↗

arXiv2025

Well-posedness of the periodic nonlinear Schr\"odinger equation with concentrated nonlinearity

We study the solution theory of the nonlinear Schr\"odinger equation with a concentrated nonlinearity on the torus. In particular, we establish existence and uniqueness of global energy-conserving solutions for initial data in $H^1$. We also prove local well-posedness of the concentrated nonlinear Schr\"odinger equation below the energy space but above the endpoint of Sobolev embedding theorem. Our methods are based on compactness results and exploiting the Volterra integral equation structure of the problem. To our knowledge, this is the first rigorous solution theory for the concentrated nonlinear Schr\"odinger equation on the circle.

math.AP↗

arXiv2025

Quantitative Derivation of the Two-Component Gross--Pitaevskii Equation in the Hard-Core Limit with Uniform-in-Time Convergence Rate

We derive the time-dependent two-component Gross--Pitaevskii (GP) equation as an effective description of the dynamics of a dilute two-component Bose gas near its ground state, which exhibits a two-component Bose-Einstein condensate, in the GP limit. Our main result establishes a uniform-in-time bound on the convergence rate between the many-body dynamics and the effective description, explicitly quantified in terms of the particle number $N$, and also implies a uniform-in-time bound for the one-component case. This improves upon the works of Michelangeli and Olgliati [77, 89] by providing a sharper, $N$-dependent, time-independent convergence rate. Our approach further extends the framework of Benedikter, de Oliveira, and Schlein [10] to the multi-component Bose gas in the hard-core limit setting. More specifically, we develop the necessary Bogoliubov theory to analyze the dynamics of multi-component Bose gases in the GP regime.

math-ph↗

arXiv2024

Derivation of the Chern-Simons-Schr\"odinger equation from the dynamics of an almost-bosonic-anyon gas

We study the time evolution of an initial product state in a system of almost-bosonic-extended-anyons in the large-particle limit. We show that the dynamics of this system can be well approximated, in finite time, by a product state evolving under the effective Chern--Simons--Schr\"odinger equation. Furthermore, we provide a convergence rate for the approximation in terms of the radius $R = (\log N)^{\frac{1}{2}+\varepsilon}$ of the extended anyons. These results establish a rigorous connection between the microscopic dynamics of almost-bosonic-anyon gases and the emergent macroscopic behavior described by the Chern--Simons--Schr\"odinger equation.

math-ph↗

arXiv2024

Exactly solvable Schr\"odinger operators related to the confluent equation

Our paper investigates one-dimensional Schr\"odinger operators defined as closed operators on $L^2(\mathbb{R})$ or $L^2(\mathbb{R}_+)$ that are exactly solvable in terms of confluent functions (or, equivalently, Whittaker functions). We allow the potentials to be complex. They fall into three families: Whittaker operators (or radial Coulomb Hamiltonians), Schr\"odinger operators with Morse potentials and isotonic oscillators. For each of them, we discuss the corresponding basic holomorphic family of closed operators and the integral kernel of their resolvents. We also describe transmutation identities that relate these resolvents. These identities interchange spectral parameters with coupling constants across different operator families. A similar analysis is performed for one-dimensional Schr\"odinger operators solvable in terms of Bessel functions (which are reducible to special cases of Whittaker functions). They fall into two families: Bessel operators and Schr\"odinger operators with exponential potentials. To make our presentation self-contained, we include a short summary of the theory of closed one-dimensional Schr\"odinger operators with singular boundary conditions. We also provide a concise review of special functions that we use.

math-ph↗

arXiv2024

Second Order Expansion of Gibbs State Reduced Density Matrices in the Gross-Pitaevskii Regime

We consider a translation-invariant system of $N$ bosons in $\mathbb{T}^{3}$ that interact through a repulsive two-body potential with scattering length of order $N^{-1}$ in the limit $N\to \infty$. We derive second order expressions for the one- and two-particle reduced density matrix matrices of the Gibbs state at fixed positive temperatures, thus obtaining a justification of Bogoliubov's prediction on the fluctuations around the condensate.

math-ph↗

arXiv2023

Derivation of Vlasov equations for multi-species fermions

We explore a system of two-species fermions with $N$ particles. Through this exploration, we establish a two-species Husimi measure and construct a two-species Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy with error terms. We provide proofs of the smallness of these errors in the semiclassical and mean-field limits, fixing $\hbar = N^{-1/3}$. After that, we rigorously establish the uniqueness of this hierarchy. This allows us to conclude that the two-species Husimi measure from the solution of the $N$-particle Schr\"odinger equation approximates the solution of the Vlasov equation for two-species systems.

math-ph↗

arXiv2023

Microscopic derivation of a Schr\"odinger equation in dimension one with a nonlinear point interaction

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 Schr\"odinger equation with a defocusing nonlinearity concentrated at a point. More precisely, we prove the 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.

math-ph↗

arXiv2023

A mixed-norm estimate of the two-particle reduced density matrix of many-body Schrödinger dynamics for deriving the Vlasov equation

We re-examine the combined semi-classical and mean-field limit in the $N$-body fermionic Schrödinger equation with pure state initial data using the Husimi measure framework. The Husimi measure equation involves three residue types: kinetic, semiclassical, and mean-field. The main result of this paper is to provide better estimates for the kinetic and mean-field residue than those in \cite{Chen2021JSP}. Especially, the estimate for the mean-field residue is shown to be smaller than the semiclassical residue by a mixed-norm estimate of the two-particle reduced density matrix factorization. Our analysis also updates the oscillation estimate parts in the residual term estimates appeared in \cite{Chen2021JSP}.

math-ph↗

arXiv2023

On the convergence of nonlinear averaging dynamics with three-body interactions on hypergraphs

Complex networked systems in fields such as physics, biology, and social sciences often involve interactions that extend beyond simple pairwise ones. Hypergraphs serve as powerful modeling tools for describing and analyzing the intricate behaviors of systems with multi-body interactions. Herein, we investigate a discrete-time nonlinear averaging dynamics with three-body interactions: an underlying hypergraph, comprising triples as hyperedges, delineates the structure of these interactions, while the vertices update their states through a weighted, state-dependent average of neighboring pairs' states. This dynamics captures reinforcing group effects, such as peer pressure, and exhibits higher-order dynamical effects resulting from a complex interplay between initial states, hypergraph topology, and nonlinearity of the update. Differently from linear averaging dynamics on graphs with two-body interactions, this model does not converge to the average of the initial states but rather induces a shift. By assuming random initial states and by making some regularity and density assumptions on the hypergraph, we prove that the dynamics converges to a multiplicatively-shifted average of the initial states, with high probability. We further characterize the shift as a function of two parameters describing the initial state and interaction strength, as well as the convergence time as a function of the hypergraph structure.

math.DS↗

arXiv2022

On the characterisation of fragmented Bose-Einstein condensation and its emergent effective evolution

Fragmented Bose-Einstein condensates are large systems of identical bosons displaying \emph{multiple} macroscopic occupations of one-body states, in a suitable sense. The quest for an effective dynamics of the fragmented condensate at the leading order in the number of particles, in analogy to the much more controlled scenario for complete condensation in one single state, is deceptive both because characterising fragmentation solely in terms of reduced density matrices is unsatisfactory and ambiguous, and because as soon as the time evolution starts the rank of the reduced marginals generically passes from finite to infinite, which is a signature of a transfer of occupations on infinitely many more one-body states. In this work we review these difficulties, we refine previous characterisations of fragmented condensates in terms of marginals, and we provide a quantitative rate of convergence to the leading effective dynamics in the double limit of infinitely many particles and infinite energy gap.

math-ph↗

arXiv2022

Uniform in Time Convergence to Bose-Einstein Condensation for a Weakly Interacting Bose Gas with an External Potential

We consider a gas of weakly interacting bosons in three dimensions subject to an external potential in the mean field regime. Assuming that the initial state of our system is a product state, we show that in the trace topology of one-body density matrices, the dynamics of the system can be described by the solution to the corresponding Hartree type equation. Using a dispersive estimate for the Hartree type equation, we obtain an error term that is uniform in time. Moreover, the dependence of the error term on the particle number is optimal.

math-ph↗

arXiv2022

Combined mean-field and semiclassical limits of large fermionic systems

We study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale $\hbar = N^{-1/3}$ in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.

math-ph↗

arXiv2021

Convergence towards the Vlasov-Poisson Equation from the $N$-Fermionic Schrödinger Equation

We consider the quantum dynamics of $N$ interacting fermions in the large $N$ limit. The particles in the system interact with each other via repulsive interaction that is regularized Coulomb potential with a polynomial cutoff with respect to $N$.From the quantum system, we derive the Vlasov-Poisson system by simultaneously estimating the semiclassical and mean-field residues in terms of the Husimi measure.

math-ph↗

arXiv2021

Rate of convergence towards mean-field evolution for weakly interacting bosons with singular three-body interactions

In this paper, we investigate the dynamics of a system of $N$ weakly interacting bosons with singular three-body interactions in three dimensions. By assuming factorized initial data $Ψ_{N,0}=φ_{0}^{\otimes N}$ and triple collisions, we prove that in the many-particle limit, its mean-field approximation converges to quintic Hartree dynamics. Moreover, we prove that the rate of convergence towards the mean-field quintic Hartree evolution is of $O(N^{-(1+4a)/(3+2a)})$ for $φ_{0}\in H^{(3/2)+a}(\mathbb{R}^{3})$ where $0\leq a<1/2$ and $O(N^{-1})$ for $a>1$. Our proof is based on and extends the Fock space approach.

math-ph↗