Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Mathematical results on many-body quantum systems

Jonas Lampart

Published Nov 5, 2024• 0 citations
OpenAlex

Abstract

This research reviewed in this habilitation thesis touches upon a variety of topics, grouped into six separate chapters. The unifying theme of all this work is a connection to the theory of many-body quantum systems. In some cases, the mathematical results are of direct relevance to such systems, like $N$-body systems or particle-field systems. In others, the connection lies in the fact that the studied equations are used as effective descriptions of many-body systems, for example in the mean-field approximation.

Related papers

Quantum Many-Particle Systems

John W. Negele, Henri Orland

This book explains the fundamental concepts and theoretical techniques used to understand the properties of quantum systems having large numbers of degrees of freedom. A number of complimentary approaches are developed, including perturbation theory; nonperturbative approximations based on functional integrals; general arguments based on order parameters, symmetry, and Fermi liquid theory; and stochastic methods.

CRC Press eBooks•Mar 5, 2018•1,402 citations

Looking at Time from a Physics Perspective

Patricio Robles

The subject of time continues to be a subject of extensive research in the development of new theories of physics. This new volume is addressed to students who are starting a graduate program in physics or electrical engineering interested in complementing their studies of relativity theory and quantum physics, applying the knowledge they have acquired about these themes to the analysis of situations where the issue of time measurement is relevant. This is the case, for example, of clock synchronization, transit times of optical signals through dielectric and absorbing media, lifetimes of excited atomic states, among others. These topics, in addition to being of great importance to theoretical physicists, are the basis of many technological developments. For example, global positional systems (GPS) are based on the predictions of relativity theory about time and the effect of gravity over time measurement. Divided into six chapters, the volume discusses how the concept of time is present in the main fields of physics, such as classical mechanics, electrodynamics, quantum mechanics and theory of relativity. Illustrative examples and case studies are included in each chapter. The volume includes an analysis of themes related to time such as causality and the arrows of time, spooky action at distance and Einstein-Podolsky-Rosen Paradox, quantum mechanics and entangled states, apparent superluminal velocity, and time reversal. This book, Looking at Time from a Physics Perspective, will contribute to the understanding of concepts learned in courses on classical mechanics, electrodynamics, quantum mechanics and relativity, reviewing the implications of the time variable for the description of the different physical phenomena at the microscopic and macroscopic level.

Apple Academic Press eBooks•Jan 29, 2024•0 citations

Brief Encounters. Binary Correlations among Bosons

Sørensen, O.

A description of a large system of particles is often sought in a derivation from the detailed behaviour of just a few of the particles. The present thesis deals with the connection between such microscopic features and the nature of a collection of many particles. The study of two-body correlations yields insight into mechanisms that in the dilute limit can be accounted for by a mean field, and yields deviations especially in the presence of a two-body bound state or a resonance. Contents: Chapter 2 presents the hyperspherical frame for studying few-body correlations within a many-boson system. Chapter 3 contains a discussion of analytical and numerical solutions to the hyperangular part of the Hamiltonian. Chapter 4 contains a discussion of the hyperradial part of the description of a many-boson system, as well as a discussion of Bose-Einstein condensation in the hyperspherical model. In chapter 5 we compare results from the previous chapters to the results from the mean-field Gross-Pitaevskii equation and ranges of validity are estimated. In chapter 6 we discuss stability criteria, approaches to the dynamical evolution of the many-boson system, and relevant time scales. Chapter 7 deals with the effects of a deformed external trap.

arXiv:cond-mat/0401323•Jan 19, 2004•1 citation

Quantum Fluctuations in the Many-Body Problem

André F. Verbeure, Valentin Anatol'evich Zagrebnov

'Quantum Fluctuations in the Many-Body Problem' published in 'Mathematical Results in Quantum Mechanics'

Operator theory•Jan 1, 1994•1 citation

Particles and quasi-particles

Richard M. Martin, Lucia Reining, David M. Ceperley

Art is the elimination of the unnecessary. Pablo Picasso Summary Condensed matter is constituted by a huge number of electrons and nuclei interacting with Coulomb potentials. The topic of this chapter is a way to deal with the full, coupled problem by separating it into a part that is tractable and the rest that is approximated. This naturally leads to the appearance of dynamical fields, and to the concept of “quasi-particles” that have the same quantum numbers as non-interacting electrons. The quasi-particles obey equations where the potential and bare interactions in the hamiltonian are replaced by dynamical self-energies and screened interactions that can describe many of the effects of correlation. In this chapter we discuss the intuitive concepts behind this approach in general terms to motivate the more rigorous formulations and approximations used in the Green's function methods of the following chapter and in Part III. The first equations of this book, Eqs. (1.1) and (1.2), express the fundamental theory of matter in terms of electrons and nuclei that interact with Coulomb potentials. For systems with only a few electrons, a solution of the problem can be obtained by exact diagonalization, using configuration interaction methods. For many-electron systems one has to resort to other methods. For example, QMC stochastic simulations (Chs. 22–25) are among the most accurate methods known to calculate certain expectation values, such as equilibrium thermodynamic properties, total energies, the density, and various static correlation functions like those in Ch. 5. Other properties such as excitation spectra are more difficult to access with QMC. Straightforward perturbation theory is, in general, not appropriate since interactions among the electrons are of the same order of magnitude as the independent-particle terms. Hence, we must develop other approaches. One strategy that gives access to equilibrium thermodynamic as well as dynamical excitation properties is to separate the interacting many-body problem into a part that is tractable by some means and the rest of the problem that is dealt with more approximately. The present chapter is devoted to this idea. It is a prelude to many-body Green's function methods; the aim is to provide a unified framework for the developments in Chs. 8–21, along with a qualitative description of the most relevant quantities.

Cambridge University Press eBooks•May 31, 2016•0 citations

Lecture Notes on Quantum Many-Body Theory: A Pedagogical Introduction

Tafuri, Fabrizio, C. A. Perroni, G. De Filippis

In these notes, we present a rigorous and self-contained introduction to the fundamental concepts and methods of quantum many-body theory. The text is designed to provide a solid theoretical foundation for the study of interacting quantum systems, combining clarity with mathematical precision. Core topics are developed systematically, with detailed derivations and comprehensive proofs that aim to make the material accessible to graduate students and beginning PhD students. Special attention is given to formal consistency and pedagogical structure, so as to guide the reader through both the conceptual and technical aspects of the subject. This work is intended as a reliable starting point for further exploration and research in modern quantum many-body physics.

arXiv:2507.09229•Jul 12, 2025•0 citations
This research reviewed in this habilitation thesis touches upon a variety of topics, grouped into six separate chapters. The unifying theme of all this work is a connection to the theory of many-body quantum systems. In some cases, the mathematical results are of direct relevance to such systems, like NNN-body systems or particle-field systems. In others, the connection lies in the fact that the studied equations are used as effective descriptions of many-body systems, for example in the mean-field approximation.
The subject of time continues to be a subject of extensive research in the development of new theories of physics. This new volume is addressed to students who are starting a graduate program in physics or electrical engineering interested in complementing their studies of relativity theory and quantum physics, applying the knowledge they have acquired about these themes to the analysis of situations where the issue of time measurement is relevant. This is the case, for example, of clock synchronization, transit times of optical signals through dielectric and absorbing media, lifetimes of excited atomic states, among others. These topics, in addition to being of great importance to theoretical physicists, are the basis of many technological developments. For example, global positional systems (GPS) are based on the predictions of relativity theory about time and the effect of gravity over time measurement. Divided into six chapters, the volume discusses how the concept of time is present in the main fields of physics, such as classical mechanics, electrodynamics, quantum mechanics and theory of relativity. Illustrative examples and case studies are included in each chapter. The volume includes an analysis of themes related to time such as causality and the arrows of time, spooky action at distance and Einstein-Podolsky-Rosen Paradox, quantum mechanics and entangled states, apparent superluminal velocity, and time reversal. This book, Looking at Time from a Physics Perspective, will contribute to the understanding of concepts learned in courses on classical mechanics, electrodynamics, quantum mechanics and relativity, reviewing the implications of the time variable for the description of the different physical phenomena at the microscopic and macroscopic level.
A description of a large system of particles is often sought in a derivation from the detailed behaviour of just a few of the particles. The present thesis deals with the connection between such microscopic features and the nature of a collection of many particles. The study of two-body correlations yields insight into mechanisms that in the dilute limit can be accounted for by a mean field, and yields deviations especially in the presence of a two-body bound state or a resonance. Contents: Chapter 2 presents the hyperspherical frame for studying few-body correlations within a many-boson system. Chapter 3 contains a discussion of analytical and numerical solutions to the hyperangular part of the Hamiltonian. Chapter 4 contains a discussion of the hyperradial part of the description of a many-boson system, as well as a discussion of Bose-Einstein condensation in the hyperspherical model. In chapter 5 we compare results from the previous chapters to the results from the mean-field Gross-Pitaevskii equation and ranges of validity are estimated. In chapter 6 we discuss stability criteria, approaches to the dynamical evolution of the many-boson system, and relevant time scales. Chapter 7 deals with the effects of a deformed external trap.
Art is the elimination of the unnecessary. Pablo Picasso Summary Condensed matter is constituted by a huge number of electrons and nuclei interacting with Coulomb potentials. The topic of this chapter is a way to deal with the full, coupled problem by separating it into a part that is tractable and the rest that is approximated. This naturally leads to the appearance of dynamical fields, and to the concept of “quasi-particles” that have the same quantum numbers as non-interacting electrons. The quasi-particles obey equations where the potential and bare interactions in the hamiltonian are replaced by dynamical self-energies and screened interactions that can describe many of the effects of correlation. In this chapter we discuss the intuitive concepts behind this approach in general terms to motivate the more rigorous formulations and approximations used in the Green's function methods of the following chapter and in Part III. The first equations of this book, Eqs. (1.1) and (1.2), express the fundamental theory of matter in terms of electrons and nuclei that interact with Coulomb potentials. For systems with only a few electrons, a solution of the problem can be obtained by exact diagonalization, using configuration interaction methods. For many-electron systems one has to resort to other methods. For example, QMC stochastic simulations (Chs. 22–25) are among the most accurate methods known to calculate certain expectation values, such as equilibrium thermodynamic properties, total energies, the density, and various static correlation functions like those in Ch. 5. Other properties such as excitation spectra are more difficult to access with QMC. Straightforward perturbation theory is, in general, not appropriate since interactions among the electrons are of the same order of magnitude as the independent-particle terms. Hence, we must develop other approaches. One strategy that gives access to equilibrium thermodynamic as well as dynamical excitation properties is to separate the interacting many-body problem into a part that is tractable by some means and the rest of the problem that is dealt with more approximately. The present chapter is devoted to this idea. It is a prelude to many-body Green's function methods; the aim is to provide a unified framework for the developments in Chs. 8–21, along with a qualitative description of the most relevant quantities.
In these notes, we present a rigorous and self-contained introduction to the fundamental concepts and methods of quantum many-body theory. The text is designed to provide a solid theoretical foundation for the study of interacting quantum systems, combining clarity with mathematical precision. Core topics are developed systematically, with detailed derivations and comprehensive proofs that aim to make the material accessible to graduate students and beginning PhD students. Special attention is given to formal consistency and pedagogical structure, so as to guide the reader through both the conceptual and technical aspects of the subject. This work is intended as a reliable starting point for further exploration and research in modern quantum many-body physics.