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Serena Cenatiempo

Publications and source records attributed to Serena Cenatiempo.

At least 19 recordsLinked to original sources
arXiv2026

The Lee-Huang-Yang energy for a dilute gas of hard spheres: an upper bound

We consider a quantum gas consisting of $N$ hard spheres with radius $\frak{a} > 0$, obeying bosonic statistics and moving in the box $Λ= [0;L]^3$ with periodic boundary conditions. We are interested in the ground state energy per unit volume in the thermodynamic limit, with $N, L \to \infty$ at fixed density $ρ= N / L^3$. We derive an upper bound for the ground state energy density, matching the famous Lee-Huang-Yang formula, up to lower order terms, in the dilute limit $ρ\frak{a}^3 \ll 1$.

math-ph↗
arXiv2025

A new upper bound on the specific free energy of dilute Bose gases

We prove an upper bound for the free energy (per unit volume) of the dilute Bose gas in the thermodynamic limit, showing that the free energy at density $ρ$ and inverse temperature $β$ differs from that of the non-interacting system by the correction term $4 π\frak{a} (2 ρ^2 - [ρ- ρ_{\textsf{c}}(β)]^2_+ )$. Here, $\frak{a}$ denotes the scattering length of the interaction potential, $ρ_{\textsf{c}}(β)$ the critical density for Bose-Einstein condensation of the non-interacting gas and $[\cdot]_+=\max\{0,\cdot\}$. This result was previously established by Yin in [37]. Our proof applies to a broader class of interaction potentials, yields a better rate, and we believe it has potential for further extensions.

math-ph↗
arXiv2023

The excitation spectrum of two dimensional Bose gases in the Gross-Pitaevskii regime

We consider a system of $N$ bosons, in the two-dimensional unit torus. We assume particles to interact through a repulsive two-body potential, with a scattering length that is exponentially small in $N$ (Gross-Pitaevskii regime). In this setting, we establish the validity of the predictions of Bogoliubov theory, determining the ground state energy of the Hamilton operator and its low-energy excitation spectrum, up to errors that vanish in the limit $N \to \infty$.

math-ph↗
arXiv2023

Upper bound for the ground state energy of a dilute Bose gas of hard spheres

We consider a gas of bosons interacting through a hard-sphere potential with radius $\frak{a}$ in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term $4πρ\frak{a}$ and shows that corrections are smaller than $C ρ\frak{a} (ρ\frak{a}^3)^{1/2}$, for a sufficiently large constant $C > 0$. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order $ρ\frak{a} (ρ\frak{a}^3)^{1/2}$, in agreement with the Lee-Huang-Yang prediction.

math-ph↗
arXiv2023

A second order upper bound for the ground state energy of a hard-sphere gas in the Gross-Pitaevskii regime

We prove an upper bound for the ground state energy of a Bose gas consisting of $N$ hard spheres with radius $\mathfrak{a}/N$, moving in the three-dimensional unit torus $Λ$. Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit $N \to \infty$. The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose-Einstein condensate and describing correlations on large scales.

math-ph↗
arXiv2022

Energy expansions for dilute Bose gases from local condensation results: a review of known results

Non-relativistic interacting bosons at zero temperature, in two and three dimensions, are expected to exhibit a fascinating critical phase, famously known as condensate phase. Even though a proof of Bose-Einstein condensation in the thermodynamic limit is still beyond reach of the current available methods, in the past decades the mathematical physics community has gained an enhanced comprehension of other aspects of the macroscopic behavior of dilute Bose gases at zero temperature. In these notes we review part of these advances, by focusing on the strict relation among the occurrence of Bose-Einstein condensation on large -- but finite -- boxes, and the asymptotic expansion to the ground state energy of dilute Bose gases.

math-ph↗
arXiv2022

Ground state energy of a Bose gas in the Gross-Pitaevskii regime

We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson's upper bound, which provides the correct leading order asymptotics for hard spheres. Afterwards, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross-Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross-Pitaeavskii limit. Here we only sketch the main ideas, details will appear elsewhere.

math-ph↗
arXiv2021

Complete Bose-Einstein condensation in the Gross-Pitaevskii regime

We consider a gas of $N$ bosons in a box with volume one interacting through a two-body potential with scattering length of order $N^{-1}$ (Gross-Pitaevskii limit). Assuming the (unscaled) potential to be sufficiently small, we show that the ground state of the system and all states with relatively small excitation energy exhibit complete Bose-Einstein condensation, with a uniform (i.e. $N$ independent) bound on the number of excitations.

math-ph↗
arXiv2021

A new second order upper bound for the ground state energy of dilute Bose gases

We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second order term, as predicted by the Lee-Huang-Yang formula. This result has been first established in J. Stat. Phys. 136(3) by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported $V \in L^3 (\mathbb{R}^3)$, gives better rates and, in our opinion, is substantially simpler.

math-ph↗
arXiv2020

Bose-Einstein condensation for two dimensional bosons in the Gross-Pitaevskii regime

We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose-Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.

math-ph↗
arXiv2019

Optimal Rate for Bose-Einstein Condensation in the Gross-Pitaevskii Regime

We consider systems of bosons trapped in a box, in the Gross-Pitaevskii regime. We show that low-energy states exhibit complete Bose-Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in \cite{BBCS1}, removing the assumption of small interaction potential.

math-ph↗
arXiv2019

Bogoliubov theory for dilute bose gases: the Gross-Pitaevskii Regime

In 1947 Bogoliubov suggested a heuristic theory to compute the excitation spectrum of weakly interacting Bose gases. Such a theory predicts a linear excitation spectrum and provides expressions for the thermodynamic functions which are believed to be correct in the dilute limit. Thus far, there are only a few cases where the predictions of Bogoliubov can be obtained by means of rigorous mathematical analysis. A major challenge is to control the corrections beyond Bogoliubov theory, namely to test the validity of Bogoliubov's predictions in regimes where the approximations made by Bogoliubov are not valid. In these notes we discuss how this challenge can be addressed in the case of a system of N interacting bosons trapped in a box with volume one in the Gross-Pitaevskii limit, where the scattering length of the potential is of the order 1/N and N tends to infinity. This is a recent result obtained in [10] and [11], joint works with C. Boccato, C. Brennecke, and B. Schlein, which extend a previous result obtained in [8], removing the assumption of small interaction potential.

math-ph↗
arXiv2018

Bogoliubov Theory in the Gross-Pitaevskii Limit

We consider Bose gases consisting of $N$ particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of the order $N^{-1}$(Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as $N \to \infty$. Our results confirm Bogoliubov's predictions.

math-ph↗
arXiv2017

Universal low-energy behavior in a quantum Lorentz gas with Gross-Pitaevskii potentials

We consider a quantum particle interacting with $N$ obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form $N^2 V(Nx)$ (Gross-Pitaevskii potential). We show convergence of the $N$ dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for $N$ large. Moreover we explicitely characterize the fluctuations around the limit operator. Our model can be considered as a simplified model for scattering of slow neutrons from condensed matter.

math-ph↗
arXiv2017

The excitation spectrum of Bose gases interacting through singular potentials

We consider systems of $N$ bosons in a box with volume one, interacting through a repulsive two-body potential of the form $κN^{3β-1} V(N^βx)$. For all $0 < β< 1$, and for sufficiently small coupling constant $κ> 0$, we establish the validity of Bogoliubov theory, identifying the ground state energy and the low-lying excitation spectrum up to errors that vanish in the limit of large $N$.

math-ph↗
arXiv2017

Analysis of fluctuations around non linear effective dynamics

We consider the derivation of effective equations approximating the many-body quantum dynamics of a large system of $N$ bosons in three dimensions, interacting through a two-body potential $N^{3β-1}V(N^βx)$. For any $0 \leq β\leq 1$ well known results establish the trace norm convergence of the k-particle reduced density matrices associated with the solution of the many-body Schrödinger equation towards products of solutions of a one-particle non linear Schrödinger equation, as $N \to \infty$. In collaboration with C. Boccato and B. Schlein we studied fluctuations around the approximate non linear Schrödinger dynamics, obtaining for all $0 < β< 1$ a norm approximation of the evolution of an appropriate class of data on the Fock space.

math-ph↗
arXiv2015

Quantum many-body fluctuations around nonlinear Schrödinger dynamics

We consider the many body quantum dynamics of systems of bosons interacting through a two-body potential $N^{3β-1} V (N^βx)$, scaling with the number of particles $N$. For $0< β< 1$, we obtain a norm-approximation of the evolution of an appropriate class of data on the Fock space. To this end, we need to correct the evolution of the condensate described by the one-particle nonlinear Schrödinger equation by means of a fluctuation dynamics, governed by a quadratic generator.

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