arXiv subjects

David Mitrouskas

Publications and source records attributed to David Mitrouskas.

23 records · Page 2Linked to original sources

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↗

arXiv2020

A note on the Fröhlich dynamics in the strong coupling limit

We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in [Gri17]. In the latter it was shown that the Fröhlich time evolution applied to the initial state $φ_0 \otimes ξ_α$, where $φ_0$ is the electron ground state of the Pekar energy functional and $ξ_α$ the associated coherent state of the phonons, can be approximated by a global phase for times small compared to $α^2$. In the present note we prove that a similar approximation holds for $t=O(α^2)$ if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to $α^{-2}$ and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order $α^2$ while the phonon fluctuations around the coherent state $ξ_α$ can be described by a time-dependent Bogoliubov transformation.

math-ph↗

arXiv2018

High Density Limit of the Fermi Polaron with Infinite Mass

We analyze the ground state energy for $N$ identical fermions in a two-dimensional box of volume $L^2$ interacting with an external point scatterer. Since the point scatterer can be considered as an impurity particle of infinite mass, this system is a limit case of the Fermi polaron. We prove that its ground state energy in the limit of high density $N/L^2 \gg 1$ 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 F. Chevy in the physics literature. The relative error in our result is shown to be small uniformly in $L$. Hence, we do not require a gap of fixed size in the spectrum of the Laplacian on the box. The strategy of our proof relies on a twofold Birman-Schwinger type argument applied to the many-particle Hamiltonian of the system.

math-ph↗

arXiv2016

Free time evolution of a tracer particle coupled to a Fermi gas in the high-density limit

The dynamics of a particle coupled to a dense and homogeneous ideal Fermi gas in two spatial dimensions is studied. We analyze the model for coupling parameter g=1 (i.e., not in the weak coupling regime), 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 which is given at leading order by the spatially homogeneous mean field potential of the gas particles. Here, the mean field approximation turns out to be accurate although the fluctuations of the potential around its mean value can be arbitrarily large. Our result is in contrast to a dense bosonic gas in which the free motion of a tracer particle would be disturbed already on a very short time scale. The proof is based on the use of strong phase cancellations in the deviations of the microscopic dynamics from the mean field time evolution.

math-ph↗

arXiv2016

Bogoliubov corrections and trace norm convergence for the Hartree dynamics

We consider the dynamics of a large number N of nonrelativistic bosons in the mean field limit for a class of interaction potentials that includes Coulomb interaction. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the N-particle space that is very similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the N-particle space. This result allows us to prove several other results: convergence of reduced density matrices in trace norm with optimal rate, convergence in energy trace norm, and convergence to a time evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected optimal rate. We thus extend and quantify several previous results, e.g., by providing the physically important convergence rates, including time-dependent external fields and singular interactions, and allowing for general initial states, e.g., those that are expected to be ground states of interacting systems.

math-ph↗