arXiv:2002.03406 · Annales Henri Poincaré

Bose–Einstein Condensation Beyond the Gross–Pitaevskii Regime

Arka Adhikari, Christian Brennecke, Benjamin Schlein

Published Dec 26, 2020• 45 citations

Abstract

Abstract We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $$N^{-1+\kappa }$$ N - 1 + κ , for $$\kappa >0$$ κ > 0 . Assuming that $$\kappa \in (0;1/43)$$ κ ∈ ( 0 ; 1 / 43 ) , we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.

DOI: 10.1007/s00023-020-01004-1

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arXiv:2002.03406 · Annales Henri Poincaré

Bose–Einstein Condensation Beyond the Gross–Pitaevskii Regime

Arka Adhikari, Christian Brennecke, Benjamin Schlein

Published Dec 26, 2020• 45 citations

arXiv HTML PDF

Abstract

Abstract We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order

N^{-1+\kappa }

N - 1 + κ , for

\kappa >0

κ > 0 . Assuming that

\kappa \in (0;1/43)

κ ∈ ( 0 ; 1 / 43 ) , we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.

DOI: 10.1007/s00023-020-01004-1

Related papers

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Cristina Caraci, Serena Cenatiempo, Benjamin Schlein

arXiv:2011.05962Journal of Statistical PhysicsMay 20, 202117 citations

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Christian Brennecke, Marco Caporaletti, Benjamin Schlein

arXiv:2104.13003Reviews in Mathematical PhysicsJun 4, 202229 citations

A Short Proof of Bose–Einstein Condensation in the Gross–Pitaevskii Regime and Beyond

Christian Brennecke, Morris Brooks, Cristina Caraci +1 more

Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order $N^{\kappa -1}$ N κ - 1 , for some $\kappa >0$ κ > 0 . For the range $\kappa \in [0, \frac{1}{43})$ κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $\kappa \in [0, \frac{1}{20})$ κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order $$ N^{\kappa -1}$$ N κ - 1 , for some $$\kappa >0$$ κ > 0 . For the range $$ \kappa \in [0, \frac{1}{43})$$ κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $$ \kappa \in [0, \frac{1}{20})$$ κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).

arXiv:2401.00784Annales Henri PoincaréJul 1, 20245 citations

Bogoliubov theory in the Gross-Pitaevskii limit

Chiara Boccato, Christian Brennecke, Serena Cenatiempo +1 more

Jan 1, 2019103 citations

A second order upper bound on the ground state energy of a Bose gas beyond the Gross–Pitaevskii regime

Giulia Basti

We consider a system of N bosons in a unitary box in the grand-canonical setting interacting through a potential with the scattering length scaling as N−1+κ, κ ∈ (0, 2/3). This regimes interpolate between the Gross–Pitaevskii regime (κ = 0) and the thermodynamic limit (κ = 2/3). In the work of Basti et al. [Forum Math., Sigma 9, E74 (2021)], as an intermediate step to prove an upper bound in agreement with the Lee–Huang–Yang formula in the thermodynamic limit, a second order upper bound on the ground state energy for κ < 5/9 was obtained. In this paper, thanks to a more careful analysis of the error terms, we extend the mentioned result to κ < 7/12.We consider a system of N bosons in a unitary box in the grand-canonical setting interacting through a potential with the scattering length scaling as N−1+κ, κ ∈ (0, 2/3). This regimes interpolate between the Gross–Pitaevskii regime (κ = 0) and the thermodynamic limit (κ = 2/3). In the work of Basti et al. [Forum Math., Sigma 9, E74 (2021)], as an intermediate step to prove an upper bound in agreement with the Lee–Huang–Yang formula in the thermodynamic limit, a second order upper bound on the ground state energy for κ < 5/9 was obtained. In this paper, thanks to a more careful analysis of the error terms, we extend the mentioned result to κ < 7/12.

arXiv:2203.01841Journal of Mathematical PhysicsJul 1, 20224 citations

Bogoliubov theory in the Gross–Pitaevskii limit

Chiara Boccato, Christian Brennecke, Serena Cenatiempo +1 more

arXiv:1801.01389Acta MathematicaJan 1, 20199 citations

Abstract We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order

N^{-1+\kappa }

N - 1 + κ , for

\kappa >0

κ > 0 . Assuming that

\kappa \in (0;1/43)

κ ∈ ( 0 ; 1 / 43 ) , we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.

Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order

N^{\kappa -1}

N κ - 1 , for some

\kappa >0

κ > 0 . For the range

\kappa \in [0, \frac{1}{43})

κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for

\kappa \in [0, \frac{1}{20})

κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).