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David Mitrouskas

Publications and source records attributed to David Mitrouskas.

At least 19 recordsLinked to original sources
arXiv2025

Adiabatic Klein-Gordon Dynamics for the Renormalized Nelson Model

We study the renormalized Nelson model in a semiclassical regime where the field becomes classical while the particle remains quantum. The degree of classicality is measured by a small parameter $\varepsilon \ll 1$. In this scaling the particle evolves on microscopic times, whereas the field exhibits nontrivial dynamics only on the macroscopic scale $t=\mathcal{O}(\varepsilon^{-2})$. The natural semiclassical model is the coupled Schr\"odinger-Klein-Gordon (SKG) system, which encodes the time-scale separation through an explicit $\varepsilon$-dependence. Based on this scale separation in SKG, we apply the adiabatic principle to derive a new PDE for the classical field, the $\varepsilon$-free adiabatic Klein-Gordon (aKG) equation, where the field is driven by the instantaneous ground state of the particle. Our main result is a norm approximation of the Nelson dynamics by the aKG solution corrected by a quasi-free fluctuation dynamics around the classical field, generated by a renormalized Bogoliubov-Nelson Hamiltonian. As a corollary, we obtain convergence of the reduced one-body densities for both subsystems, where the fluctuation correction vanishes, thereby justifying aKG as a semiclassical Born-Oppenheimer type approximation of the renormalized Nelson model.

math-ph↗
arXiv2025

Derivation of the time-dependent Hartree equations for strongly interacting dense fermionic systems

The time-dependent Hartree and Hartree-Fock equations provide effective mean-field descriptions for the dynamics of large fermionic systems and play a fundamental role in many areas of physics. In this work, we rigorously derive the time-dependent Hartree equations as the large-$N$ limit of the microscopic Schr\"odinger dynamics of $N$ fermions confined to a volume of order one and interacting via strong pair potentials. A central step in our analysis is the implementation of time-dependent gauge transformations, which eliminate the dominant contribution from the interaction potential in both the Schr\"odinger and Hartree evolutions.

math-ph↗
arXiv2025

Bound excited states of Fr\"ohlich polarons in one dimension

The one-dimensional Fr\"ohlich model describing the motion of a single electron interacting with optical phonons is a paradigmatic model of quantum many-body physics. We predict the existence of an arbitrarily large number of bound excited states in the strong coupling limit and calculate their excitation energies. Numerical simulations of a discretized model demonstrate the complete amelioration of the projector Monte Carlo sign problem by walker annihilation in an infinite Hilbert space. They reveal the threshold for the occurrence of the first bound excited states at a value of $\alpha \approx 1.73$ for the dimensionless coupling constant. This puts the threshold into the regime of intermediate interaction strength. We find a significant spectral weight and increased phonon number of the bound excited state at threshold.

math-ph↗
arXiv2025

Emergence of fermion-mediated interactions in Bose-Fermi mixtures

This work is inspired by recent experimental observations in ultracold atomic Bose-Fermi mixtures [DeSalvo et al., Nature 568 (2019)]. These experiments reveal the emergence of an attractive fermion-mediated interaction between bosons, as well as a stability-instability transition. We give the first mathematical demonstration of this transition by studying the low-energy spectrum of a many-body interspecies Hamiltonian. More precisely, we show the convergence of its eigenvalues towards those of an effective Bose Hamiltonian, which includes fermion-mediated effects. Applying this result to a model with short-range potentials, we derive a stability-instability transition in the bosonic subsystem, driven by the Bose-Fermi coupling strength $g$. For small $|g|$, the bosons form a stable Bose-Einstein condensate with the energy per particle uniformly bounded from below. For large $|g|$, the energy per particle is no longer uniformly bounded from below, signaling the collapse of the condensate.

math-ph↗
arXiv2024

The renormalized Nelson model in the weak coupling limit

The Nelson model describes non-relativistic particles coupled to a relativistic Bose scalar field. In this article, we study the renormalized version of the Nelson model with massless bosons in Davies' weak coupling limit. Our main result states that the two-body Coulomb potential emerges as an effective pair interaction between the particles, which arises from the exchange of virtual excitations of the quantum field.

math-ph↗
arXiv2024

Radiative corrections to the dynamics of a tracer particle coupled to a Bose scalar field

We consider a tracer particle coupled to a Bose scalar field and study the regime where the field's propagation speed approaches infinity. For initial states devoid of field excitations, we introduce an effective approximation of the time-evolved wave function and prove its validity in Hilbert space norm. In this approximation, the field remains in the vacuum state while the tracer particle propagates with a modified dispersion relation. Physically, the new dispersion relation can be understood as the effect of radiative corrections due to interactions with virtual bosons. Mathematically, it is defined as the solution of a self-consistent equation, whose form depends on the relevant time scale.

math-ph↗
arXiv2023

A Note on the Binding Energy for Bosons in the Mean-field Limit

We consider a gas of N weakly interacting bosons in the ground state. Such gases exhibit Bose-Einstein condensation. The binding energy is defined as the energy it takes to remove one particle from the gas. In this article, we prove an asymptotic expansion for the binding energy, and compute the first orders explicitly for the homogeneous gas. Our result addresses in particular a conjecture by Nam [Lett. Math. Phys., 108(1):141--159, 2018], and provides an asymptotic expansion of the ionization energy of bosonic atoms.

math-ph↗
arXiv2023

Renormalized Bogoliubov Theory for the Nelson Model

We consider the time evolution of the renormalized Nelson model, which describes $N$ bosons linearly coupled to a quantized scalar field, in the mean-field limit of many particles $N\gg 1$ with coupling constant proportional to $N^{-1/2}$. First, we show that initial states exhibiting Bose-Einstein condensation for the particles and approximating a coherent state for the quantum field retain their structure under the many-body time evolution. Concretely, the dynamics of the reduced densities are approximated by solutions of two coupled PDEs, the Schr\"odinger-Klein-Gordon equations. Second, we construct a renormalized Bogoliubov evolution that describes the quantum fluctuations around the Schr\"odinger-Klein-Gordon equations. This evolution is used to extend the approximation of the evolved many-body state to the full norm topology. In summary, we provide a comprehensive analysis of the Nelson model that reveals the role of renormalization in the mean-field Bogoliubov theory.

math-ph↗
arXiv2023

Asymptotic analysis of the weakly interacting Bose gas: A collection of recent results and applications

We consider a gas of N bosons with interactions in the mean-field scaling regime. We review a recent proof of the asymptotic expansion of its spectrum and eigenstates and two applications of this result, namely the derivation of an Edgeworth expansion for fluctuations of one-body operators and the computation of the binding energy of an inhomogeneous Bose gas to any order. Finally, we collect related results for the dynamics of the weakly interacting Bose gas and for the regularized Nelson model.

math-ph↗
arXiv2022

Ubiquity of bound states for the strongly coupled polaron

We study the spectrum of the Fr\"ohlich Hamiltonian for the polaron at fixed total momentum. We prove the existence of excited eigenvalues between the ground state energy and the essential spectrum at strong coupling. In fact, our main result shows that the number of excited energy bands diverges in the strong coupling limit. To prove this we derive upper bounds for the min-max values of the corresponding fiber Hamiltonians and compare them with the bottom of the essential spectrum, a lower bound on which was recently obtained in [1]. The upper bounds are given in terms of the ground state energy band shifted by momentum-independent excitation energies determined by an effective Hamiltonian of Bogoliubov-type.

math-ph↗
arXiv2022

On the global minimum of the energy-momentum relation for the polaron

For the Fr\"ohlich model of the large polaron, we prove that the ground state energy as a function of the total momentum has a unique global minimum at momentum zero. This implies the non-existence of a ground state of the Fr\"ohlich Hamiltonian and thus excludes the possibility of a localization transition at finite coupling.

math-ph↗
arXiv2022

Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron

We consider the large polaron described by the Fr\"ohlich Hamiltonian and study its energy-momentum relation defined as the lowest possible energy as a function of the total momentum. Using a suitable family of trial states, we derive an optimal parabolic upper bound for the energy-momentum relation in the limit of strong coupling. The upper bound consists of a momentum independent term that agrees with the predicted two-term expansion for the ground state energy of the strongly coupled polaron at rest, and a term that is quadratic in the momentum with coefficient given by the inverse of twice the classical effective mass introduced by Landau and Pekar.

math-ph↗
arXiv2021

Bogoliubov Dynamics and Higher-order Corrections for the Regularized Nelson Model

We study the time evolution of the Nelson model in a mean-field limit in which N non-relativistic bosons weakly couple (w.r.t. the particle number) to a positive or zero mass quantized scalar field. Our main result is the derivation of the Bogoliubov dynamics and higher-order corrections. More precisely, we prove the convergence of the approximate wave function to the many-body wave function in norm, with a convergence rate proportional to the number of corrections taken into account in the approximation. We prove an analogous result for the unitary propagator. As an application, we derive a simple system of PDEs describing the time evolution of the first- and second-order approximation to the one-particle reduced density matrices of the particles and the quantum field, respectively.

math-ph↗
arXiv2021

Effective pair interaction between impurity particles induced by a dense Fermi gas

We study the dynamics of a small number of impurity particles coupled to the ideal Fermi gas in a $d$-dimensional box. The impurities interact with the fermions via a two-body potential $λv(x)$ where $λ$ is a coupling constant and $v(x)$ for instance a screened Coulomb potential. After taking the large-volume limit at positive Fermi momentum $k_F$ we consider the regime of high density of the fermions, that is, $k_F$ large compared to one. For coupling constants that scale like $λ^2 \sim k_F^{(2-d)}$ we show that the impurity particles effectively decouple from the fermions but evolve with an attractive pair interaction among each other which is induced by fluctuations in the Fermi gas.

math-ph↗
arXiv2021

Derivation of the Landau-Pekar equations in a many-body mean-field limit

We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle number and suitably small coupling, we show that the dynamics of the system is approximately described by the Landau-Pekar equations. These describe a Bose-Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order.

math-ph↗
arXiv2020

Weak coupling limit for the ground state energy of the 2D Fermi polaron

We analyze the ground state energy for N fermions in a two-dimensional box interacting with an impurity particle via two-body point interactions. We allow for mass ratios M > 1.225 between the impurity mass and the mass of a fermion and consider arbitrarily large box sizes while keeping the Fermi energy fixed. Our main result shows that the ground state energy in the limit of weak coupling is given by the polaron energy. The polaron energy is an energy estimate based on trial states up to first order in particle-hole expansion, which was proposed by Chevy in the physics literature. For the proof we apply a Birman-Schwinger principle that was recently obtained by Griesemer and Linden. One main new ingredient is a suitable localization of the polaron energy.

math-ph↗