Exponential Control of Excitations for Trapped BEC in the Gross-Pitaevskii Regime

Christian Brennecke; Nils Behrmann; Simone Rademacher

REVIEW 3 major objections 6 minor 33 references

Exponential Control of Excitations for Trapped BEC in the Gross-Pitaevskii Regime

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A trapped dilute Bose gas has exponentially small probability of large condensate depletion in low-energy states.

desk verdict Trapped BEC exponential control is a credible and clearly explained generalization of the torus result, but the load-bearing commutator estimates are deferred in the text, so the main theorem is not fully established from this manuscript. read the letter →

arxiv 2501.15803 v1 pith:347RCSRZ submitted 2025-01-27 math-ph math.MP

Nils Behrmann , Christian Brennecke , Simone Rademacher This is my paper · ORCID

classification math-phmath.MP MSC 35Q5581Q1082B10

keywords Bose-EinsteincondensationGross-PitaevskiiregimeexcitationnumberoperatorexponentialmomentboundgeneralizedBogoliubovtransformationlargedeviationstrappedBosegaslow-energyspectralsubspace

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves exponential control of the number of excited particles in a trapped three-dimensional Bose gas in the Gross-Pitaevskii regime: for any low-energy state, the expectation value of $\exp(\kappa N_{\perp\phi_{\mathrm{GP}}})$ is bounded uniformly in the particle number $N$, provided $\kappa$ is small enough. By Markov's inequality this means the probability of finding at least $n$ particles outside the condensate decays like $e^{-\kappa n}$, which is much stronger than the polynomial decay that follows from earlier moment bounds. The result extends to trapped gases a theorem that was known for translation-invariant gases on a torus, bringing the exponential bound closer to experimental setups with an external trap. The proof works directly on spectral subspaces of the full Hamiltonian, not only on eigenstates, and its core is a commutator estimate for a renormalized excitation Hamiltonian.

What carries the argument

The mechanism is a renormalized excitation Hamiltonian built from a generalized Bogoliubov transformation. The transformation $e^{B(\eta)}$ is generated by a two-particle correlation kernel $\eta$ that encodes the short-scale scattering correlations between particles and is constructed from the scattering correction $w_\ell$, a high-momentum cut-off, and projections orthogonal to the condensate. Conjugating the excitation Hamiltonian $L_N$ with $e^{B(\eta)}$ gives $G_N$, in which dangerous terms cancel and the relevant object is the excitation number operator $\mathcal{N}$, unitarily equivalent to $N_{\perp\phi_{\mathrm{GP}}}$. The load-bearing estimate is the commutator bound (3.8): for small $\kappa$, $|\langle\xi, e^{\kappa\mathcal{N}/2}[G_N,e^{\kappa\mathcal{N}/2}]\xi\rangle| \le C\kappa \langle\xi,(\mathcal{H}_N+\mathcal{N})e^{\kappa\mathcal{N}}\xi\rangle$, with $\mathcal{H}_N$ a renormalized kinetic-plus-interaction operator. This bound, combined with the lower bound $G_N \ge N E_{\mathrm{GP}}(\phi_{\mathrm{GP}})+c(\mathcal{H}_N+\mathcal{N})-C$ and a bootstrap comparing $\langle e^{\kappa\mathcal{N}}\rangle$ to its derivative, yields the uniform exponential moment bound. The same commutator bootstrap is first illustrated in the simpler mean-field translation-invariant setting, where the computation is explicit.

What would settle it

Write out the proof of Lemma 4.2 for the kernel $\eta$ of (3.4) and check the claimed bounds on $[e^{\lambda\mathcal{N}}, d_{\eta,x}]$ and on the double commutators; if any of these bounds fails for a potential satisfying Assumption 1.1, the commutator estimate (3.8) and with it the proof of Theorem 1.2 breaks. A less structural check is to compute, for a sequence of $N$, the exponential moment $\langle \psi_N, e^{\kappa N_{\perp\phi_{\mathrm{GP}}}}\psi_N\rangle$ for low-energy states of a trapped gas and look for divergence.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: under mild assumptions on the nonnegative interaction $V$ and the trapping potential $V_{\mathrm{ext}}$, for any fixed energy window $\zeta$ above the ground state energy $E_N$, every normalized $N$-particle wave function $\psi_N$ with energy at most $E_N+\zeta$ satisfies $\langle\psi_N,\exp(\kappa N_{\perp\phi_{\mathrm{GP}}})\psi_N\rangle \le C$ for all sufficiently small $\kappa>0$, with $C$ independent of $N$ and $\psi_N$. Here $\phi_{\mathrm{GP}}$ is the unique positive minimizer of the Gross-Pitaevskii functional and $N_{\perp\phi_{\mathrm{GP}}}$ counts particles orthogonal to the condensate wave function. The analogue on the unit torus was proved in [27]; this paper adapts the argument to the position-space setting of a trap, where momentum space is not discrete. The exponential moment bound yields the tail estimate $\mathbb{P}[N_{\perp\phi_{\mathrm{GP}}} \ge n] \le e^{-\kappa n}$ for low-energy states, so the depletion of the condensate has subexponential probability of being large. The paper also provides an illustrative mean-field version of the argument and states that it expects a full moment-generating-function and large-deviation refinement in the trapped setting.

Load-bearing premise

The load-bearing premise is that the commutator error estimates for the renormalized Hamiltonian, which the proof takes over from the translation-invariant torus setting 'up to minor technical modifications,' remain valid in position space on $\mathbb{R}^3$; the paper asserts this transfer in Lemma 4.2 and does not derive it.

Editorial extensions

If this is right

For any low-energy state of a trapped Gross-Pitaevskii gas, the probability of finding at least $n$ particles outside the condensate decays at least as $e^{-\kappa n}$, uniformly in the particle number $N$.
Expanding the exponential shows the uniform moment bound $\langle \psi_N, N_{\perp\phi_{\mathrm{GP}}}^k \psi_N\rangle \le C_k$ for every fixed $k$, recovering and strengthening the optimal-rate condensation estimates of [11,12,25].
The bound holds on the whole spectral subspace of energies up to $E_N+\zeta$, so it applies to approximate ground states and low-energy ensembles, not only to individual eigenfunctions.
The mean-field analogue proved in Section 2 shows the same commutator bootstrap works without the correlation-kernel renormalization, isolating the role of the renormalization in the Gross-Pitaevskii regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof's only deferred step is the position-space transfer of [27, Lemma 2.4] into Lemma 4.2; a reader extending this work should verify that transfer first, since a failure there would not disprove exponential control but would invalidate this proof.
A natural next step, which the paper leaves open, is to derive an explicit limiting moment generating function for $N_{\perp\phi_{\mathrm{GP}}}$ in the trap, following the torus route of [31] using [12,28].
Because the commutator estimate is formulated on Fock space and does not use stationarity, the same strategy may yield exponential control of the depletion along the Gross-Pitaevskii time evolution, not just in equilibrium.
The constant $\kappa$ in Theorem 1.2 is not made quantitative; computing its dependence on the trap and the energy window $\zeta$ would be a concrete numerical test of the bound.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper claims exponential control of the number of excitations for trapped BECs in the Gross-Pitaevskii regime, generalizing [27] from the torus to R^3 with an external potential. If the main theorem holds, it is a natural and useful extension: it gives the exponential moment bound that feeds into large-deviation and moment-generating-function results. The statement itself is new, and the high-level strategy is coherent. The mean-field analogue in Section 2 is proved completely and is a clean illustration of the commutator method. The reduction of Theorem 1.2 to Proposition 3.2 is also clearly laid out. The paper is honest about what it does not prove: Remark 1.3 mentions the master thesis [4], and the text repeatedly states 'up to minor technical modifications' and 'we omit the details'.

The soft spot is exactly this: the load-bearing technical core is deferred. Lemma 4.2 transfers the commutator error bounds of [27, Lemma 2.4] from the torus (discrete momentum space) to position space R^3 for the kernel eta in (3.4). That kernel is not translation-invariant; it contains Q x Q acting on phi_GP x phi_GP times a radial function, with phi_GP decaying but spatially varying. On the torus, Fourier modes decouple and the discrete momentum bounds simplify the double-commutator estimates. In R^3, derivatives hitting phi_GP and the lack of compactness could introduce extra terms. The paper asserts these are 'minor technical modifications' and gives no derivation. Propositions 4.4, 4.5, and 4.6, which control the errors E_N^(1) through E_N^(4) under the exponential weight, are likewise asserted by reference to [11,12,27] without proof. These are not peripheral estimates; the commutator bound (3.8) of Proposition 3.2 depends on them directly, and without (3.8), Theorem 1.2 does not follow.

To be clear: I see no evidence that the bounds are false, and the paper's own structure makes the dependency explicit. The author overlap with [11,12,27] is not a problem in itself; the new statement is clearly theirs. But a rigorous proof needs either a supplement spelling out Lemma 4.2 and the propositions, or precise pointers to where each statement appears verbatim in the cited works. As it stands, the paper is a credible research announcement with a proof sketch, not a complete proof.

Who is this for? Researchers working on rigorous BEC theory and large deviations for Bose gases. They will want to read it, but they should treat the main theorem as conditional pending the deferred estimates. I would send it to a serious referee, with instructions to check whether the claimed 'minor modifications' are indeed minor, especially for the R^3 kernel. If the referee can fill the gaps, the result is a solid contribution. If not, the paper needs substantial revision.

Recommendation: engage with the paper, but require the technical details before accepting the theorem as established.

Referee Report

3 major / 6 minor

Summary. The paper treats N trapped bosons in R^3 interacting via N^2 V(N(x-y)) and proves an exponential bound for the number of excitations out of the Gross-Pitaevskii minimizer: for low-energy spectral subspaces, ⟨ψ_N, exp(κ N_{⊥φ_GP}) ψ_N⟩ ≤ C uniformly in N (Theorem 1.2). The proof proceeds in three stages. Section 2 gives a complete proof of the analogous exponential bound for translation-invariant mean-field bosons on the torus. Section 3 reduces the Gross-Pitaevskii problem to Proposition 3.2: after conjugating the excitation Hamiltonian by a generalized Bogoliubov transformation generated by the kernel η in (3.4), the renormalized Hamiltonian G_N satisfies an energy lower bound (3.7) and an exponential commutator bound (3.8). The main theorem follows from these bounds by a Grönwall argument. Section 4 analyzes G_N term by term, but several decisive estimates (Lemma 4.2 and Propositions 4.4–4.6) are asserted with references to [11, 12, 27] and 'minor technical modifications' rather than proved.

Significance. If the proof is completed, Theorem 1.2 is a natural and valuable extension of [27] from translation-invariant settings to trapped gases, and it strengthens the O(1) moment bounds of [11, 12] to exponential moment control. The paper is cleanly organized, gives a self-contained mean-field model, identifies the external inputs (the lower bound (1.8), the energy asymptotics, and the kernel bounds of Lemma 3.1), and does not appear to assume the conclusion. The main obstacle is that the technical core supporting Proposition 3.2 is deferred; as it stands the manuscript does not fully establish its central claim.

major comments (3)

[4.1, Lemma 4.2] Lemma 4.2 is the principal technical input for the commutator bound (3.8), but its proof is not given: the text states that [27, Lemma 2.4] applies in position space on R^3 'up to minor technical modifications' and then 'we skip the details.' This is load-bearing because the kernel η in (3.4) is not translation-invariant: it contains Q⊗Q and the spatially varying, exponentially decaying φ_GP, so the discrete-momentum arguments of [27] do not directly transfer; one must control commutators in which derivatives hit φ_GP and the kernel η, as well as boundary terms at infinity. I ask for a complete proof, or a precise theorem statement in a published reference with hypotheses that match the present setting, before (3.8) can be used.
[4.3–4.5, Propositions 4.4, 4.5, 4.6] The estimates for G_N^(1) through G_N^(4) are asserted rather than proved: Proposition 4.4 is justified by 'we omit further details and refer to [11, Section 4.2] ... [27, Section 3.2]', Proposition 4.5 by 'we omit the details here', and Proposition 4.6 by 'We again skip the details'. These propositions control both the expectation values and the exponential commutator bounds for the error terms E_N^(1)–E_N^(4); they feed directly into (4.56)–(4.57) and hence into the lower bound (3.7) and the exponential bound (3.8). Since the trapping potential, the kernel η, and the generalized Bogoliubov transformation differ from the torus setting, the asserted 'same arguments' need to be written out or cited to a theorem that covers this exact configuration.
[4.6, around Eq. (4.58)] In the proof of Proposition 3.2, the scattering term displayed in (4.58) is claimed to be bounded by C(N_{⊥φ_GP}+1) with no derivation. This bound is needed for the energy lower bound (3.7), which is itself used in the proof of Theorem 1.2. Please provide the argument (including the treatment of the boundary terms from the cutoff at |x−y|=ℓ), or give a precise reference to a statement with the same trapping setup.

minor comments (6)

[Abstract, Remark 1.5] There are typos in the abstract ('Gross-Pit aevskii', 'Bose-Einst ein') and in Remark 1.5 ('everty 0 ≤ n ≤ N' should be 'every 0 ≤ n ≤ N').
[Eqs. (4.8), (4.12), (4.13)] The parentheses in `(N + 1)n+6)/2` are unbalanced, and in (4.8) the term `‖η‖2‖ay‖(N + 1)(n+4)/2ξ‖` appears to be missing a norm around `ay(N+1)^{(n+4)/2}ξ`; please check all occurrences.
[Eq. (4.39) and Section 4.5] The function in (4.39) is written as `ω(N·)`, but the function defined earlier is `wℓ = 1 − fℓ`; Section 4.5 uses `w(N·)`. Please unify the notation.
[Proof of Theorem 2.1] In the derivation of (2.3), the vector `(H_N^mf − E_N^mf)ψ_N / ||(H_N^mf − E_N^mf)ψ_N||` is placed in Q_ζ. If the denominator vanishes, this step should be justified by a limiting argument or by treating ground states separately.
[Section 4.1] The sentence 'The proof of Lemma 4.2 goes bac to NR' is an incomplete attribution; please give the full reference or remove it.
[Eq. (4.49)] The parentheses in the displayed formula for G_N^(3) are mismatched; it should read `−√N [b(coshη(h_N)) + b∗(sinhη(h_N)) + h.c.]`.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.2 is derived from external published inputs, not from the target bound.

full rationale

The derivation of Theorem 1.2 does not assume the exponential bound it proves. The proof uses the lower bound (1.8) from [11, Theorem 1.1], the energy asymptotics E_N = N E_GP(phi_GP) + O(1) from [25, 11, 28, 12], and the commutator machinery of [27, Lemma 2.4] as adapted to position space; none of these inputs contains the trapped-system exponential bound. Although [11], [12], and [27] share authors with the present paper, they are prior published or in-press results with parameter-free assumptions that do not include the target statement, so under the independence rule they count as genuine evidence rather than circular self-reference. The manuscript does defer several technical estimates, notably Lemma 4.2 and Propositions 4.4-4.6, saying that the torus arguments apply 'up to minor technical modifications' and omitting details; this is an incompleteness or rigor risk, but it is not an equivalence-by-construction of inputs and outputs. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The theorem is a parameter-free a priori bound with no fitted values. The input material consists of published results on the GP minimizer, scattering theory, and the low-energy lower bound (1.8), plus four technical statements whose proofs are deferred to prior work. The only ad hoc element is the family of deferred estimates; everything else is standard, cited mathematics.

assumptions (4)

domain assumption Assumption 1.1: V ∈ L^3(R^3) is radial, compactly supported and nonnegative; Vext ∈ C^1 is confining, ∇Vext has at most exponential growth, and Vext satisfies the product-type bound Vext(x+y) ≤ C(Vext(x)+C)(Vext(y)+C).
Enters through the lower bound (1.8) from [11, Theorem 1.1], through the definition of the excitation Fock space, and through the regularity and decay of the GP minimizer quoted in Theorem A.1.
standard math Existence, uniqueness, strict positivity and exponential decay of the Gross-Pitaevskii minimizer φ_GP (Theorem A.1, quoting [24] and [11, Appendix A]).
Used in Section 3 to define the condensation projector Q, the truncated Fock space, the kernel η in (3.4), and the pointwise bounds |η(x,y)| ≤ C N |φ_GP(x)||φ_GP(y)| of Lemma 3.1.
standard math Properties of the zero-energy scattering solution: f_ℓ solves (-Δ + V/2) f_ℓ = λ_ℓ f_ℓ on a ball, the rescaled version solves the N-dependent equation, and w_ℓ = 1 - f_ℓ satisfies the decay bounds of [11, Lemma 3.1].
Used to define the correlation kernel η in (3.4) and to prove Lemma 3.1, in particular the smallness bound ||η|| ≤ C ℓ^{α/2} for small ℓ.
ad hoc to paper Lemma 4.2 and Propositions 4.4, 4.5, 4.6 are asserted without proof: position-space commutator bounds for e^{λN} with d_{η,x}, and decompositions of the renormalized Hamiltonian terms G_N^{(1)} through G_N^{(4)}.
The paper states these follow from [27, Lemma 2.4] and the arguments of [11, 12] 'up to minor technical modifications' and explicitly omits the derivations. They are load-bearing for the commutator bound (3.8) and the lower bound (3.7) in Proposition 3.2.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Exponential Control of Excitations for Trapped BEC in the Gross-Pitaevskii Regime." pith.science (2026). https://pith.science/paper/347RCSRZ

@misc{pith2026250115803,
  author       = {Pith},
  title        = {Pith review of: Exponential Control of Excitations for Trapped BEC in the Gross-Pitaevskii Regime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/347RCSRZ}},
  note         = {Machine review of arXiv:2501.15803}
}

read the original abstract

We consider trapped Bose gases in three dimensions in the Gross-Pitaevskii regime whose low energy states are well known to exhibit Bose-Einstein condensation. That is, the majority of the particles occupies the same condensate state. We prove exponential control of the number of particles orthogonal to the condensate state, generalizing recent results for translation invariant systems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 32 canonical work pages

[27]

P. T. Nam, S. Rademacher, Exponential bounds of the cond ensation for dilute Bose gases. Trans. Amer. Math. Soc. , in press. Preprint: arXiv:2307.10622 (2023)

work page Pith review arXiv 2023
[4]

Behrmann

N. Behrmann. Exponential control of excitations in the G ross-Pitaevskii regime. Master Thesis , Institute for Applied Mathematics, University of Bonn (20 24)

work page
[1]

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Observation of Bose-Einstein condensation in a dilute atom ic vapor. Science 269, 198-201, (1995)

work page 1995
[2]

Adhikari, C

A. Adhikari, C. Brennecke, B. Schlein. Bose-Einstein Co ndensation Beyond the Gross-Pitaevskii Regime. Ann. Henri Poincar´ e22, 1163-1233 (2021)

work page 2021
[3]

Ben Arous, K

G. Ben Arous, K. Kirkpatrick, B. Schlein. Central Limit T heorem in Many-Body Quantum Dynamics. Commun. Math. Phys , 321, pp. 371-417 (2013)

work page 2013
[5]

Boccato, C

C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. Com plete Bose-Einstein con- densation in the Gross-Pitaevskii regime. Comm. Math. Phys. 359, no. 3, 975-1026 (2018)

work page 2018
[6]

Boccato, C

C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. The excitation spectrum of Bose gases interacting through singular potentials. J. Eur. Math. Soc. 22, no. 7 (2020)

work page 2020
[7]

Boccato, C

C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. Bog oliubov Theory in the Gross-Pitaevskii limit. Acta Mathematica 222, no. 2, 219-335 (2019)

work page 2019

Show all 33 references

[8]

Boccato, C

C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. Opt imal Rate for Bose- Einstein Condensation in the Gross-Pitaevskii Regime. Comm. Math. Phys. 376, 1311-1395 (2020)

2020
[9]

Brennecke, W

C. Brennecke, W. Kroschinsky. Derivation of the Gross-P itaevskii Dynamics through Renormalized Excitation Number Operators. Prepri nt: arXiv:2407.10650

arXiv
[10]

Brennecke, B

C. Brennecke, B. Schlein. Gross-Pitaevskii dynamics f or Bose-Einstein condensates. Analysis & PDE 12, no. 6, 1513-1596 (2019)

2019
[11]

Brennecke, B

C. Brennecke, B. Schlein, S. Schraven. Bose-Einstein C ondensation with Optimal Rate for Trapped Bosons in the Gross-Pitaevskii Regime. Math. Phys. Anal. Geom. 25, 12. (2022)

2022
[12]

Brennecke, B

C. Brennecke, B. Schlein, S. Schraven. Bogoliubov Theo ry for Trapped Bosons in the Gross-Pitaevskii Regime. Ann. Henri Poincar´ e23, 1583–1658 (2022)

2022
[13]

S. Bose. Plancks Gesetz und Lichtquantenhypothese. Z. Phys. 26, 178–181, (1924)

1924
[14]

M. Brooks. Diagonalizing Bose Gases in the Gross-Pitae vskii Regime and Beyond. Preprint: arXiv:2310.11347. 26

arXiv
[15]

Caraci, J

C. Caraci, J. Oldenburg and B. Schlein, Quantum Fluctua tions of Many-Body Dy- namics around the Gross-Pitaevskii Equation, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire (2024)

2024
[16]

Einstein

A. Einstein. Quantentheorie des einatomigen idealen G ases. Sitzungsberichte der Preußischen Akademie der Wissenschaften XXII, 261–267, (1924)

1924
[17]

Hainzl, B

C. Hainzl, B. Schlein and A. Triay. Bogoliubov theory in the Gross-Pitaevskii limit: a simpliﬁed approach. Forum Math. Sigma 10 (e10) , (2022)

2022
[18]

Mitrouskas

D. Mitrouskas. Derivation of mean-ﬁeld equations and n ext-order corrections for bosons and fermions. Dissertation, LMU Munich (2017)

2017
[19]

Mitrouskas, P

D. Mitrouskas, P. Pickl. Exponential decay of the numbe r of excitations in the weakly interacting Bose gas. J. Math. Phys. 64, 121901 (2023)

2023
[20]

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D . S. Durfee, D. M. Kurn, and W. Ketterle. Bose-Einstein Condensation in a Gas o f Sodium Atoms. Phys. Rev. Lett. 75, 3969–3973, (1995)

1995
[21]

Lewin, P

M. Lewin, P. T. Nam, S. Serfaty, J.P. Solovej. Bogoliubo v spectrum of interacting Bose gases. Comm. Pure Appl. Math. 68 (2014), 3, 413 - 471

2014
[22]

E. H. Lieb, R. Seiringer. Proof of Bose-Einstein conden sation for dilute trapped gases. Phys. Rev. Lett. 88, 170409 (2002)

2002
[23]

E. H. Lieb, R. Seiringer. Derivation of the Gross-Pitae vskii equation for rotating Bose gases. Comm. Math. Phys. 264:2, 505-537 (2006)

2006
[24]

E. H. Lieb, R. Seiringer, J. Yngvason. Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev. A 61 (2000)

2000
[25]

P. T. Nam, M. Napi´ orkowski, J. Ricaud, A. Triay. Optima l rate of condensation for trapped bosons in the Gross-Pitaevskii regime. Analysis & PDE 15, no. 6, 1585- 1616 (2022)

2022
[26]

P. T. Nam, N. Rougerie, R. Seiringer. Ground states of la rge bosonic systems: The Gross-Pitaevskii limit revisited. Analysis and PDE. 9, no. 2, 459-485 (2016)

2016
[28]

P. T. Nam, A. Triay. Bogoliubov excitation spectrum of t rapped Bose gases in the Gross-Pitaevskii regime. J. Math. Pures Appl. 176, 18-101 (2023)

2023
[29]

Portinale, S

L. Portinale, S. Rademacher, D. Virosztek, Limit theor ems for empirical measures of interacting quantum systems in Wasserstein space. Prepr int: arXiv:2312.0054 (2023). 27

2023
[30]

Rademacher

S. Rademacher. Large deviations for the ground state of weakly interacting Bose gases. Ann. Henri Poincar´ e(2024)

2024
[31]

Rademacher, Generating function for quantum deplet ion of Bose-Einstein con- densates

S. Rademacher, Generating function for quantum deplet ion of Bose-Einstein con- densates. Preprint: arXiv:2410.13576 (2024)

2024 arXiv
[32]

Rademacher, B

S. Rademacher, B. Schlein. Central limit theorem for Bo se-Einstein condensates. J. Math. Phys. 60 (7):071902 (2019)

2019
[33]

Seiringer

R. Seiringer. The Excitation Spectrum for Weakly Inter acting Bosons. Comm. Math. Phys. 306, no. 2, pp. 565-578 (2011). 28

2011

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [27]

P. T. Nam, S. Rademacher, Exponential bounds of the cond ensation for dilute Bose gases. Trans. Amer. Math. Soc. , in press. Preprint: arXiv:2307.10622 (2023)

work page Pith review arXiv 2023

[2] [4]

Behrmann

N. Behrmann. Exponential control of excitations in the G ross-Pitaevskii regime. Master Thesis , Institute for Applied Mathematics, University of Bonn (20 24)

work page

[3] [1]

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Observation of Bose-Einstein condensation in a dilute atom ic vapor. Science 269, 198-201, (1995)

work page 1995

[4] [2]

Adhikari, C

A. Adhikari, C. Brennecke, B. Schlein. Bose-Einstein Co ndensation Beyond the Gross-Pitaevskii Regime. Ann. Henri Poincar´ e22, 1163-1233 (2021)

work page 2021

[5] [3]

Ben Arous, K

G. Ben Arous, K. Kirkpatrick, B. Schlein. Central Limit T heorem in Many-Body Quantum Dynamics. Commun. Math. Phys , 321, pp. 371-417 (2013)

work page 2013

[6] [5]

Boccato, C

C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. Com plete Bose-Einstein con- densation in the Gross-Pitaevskii regime. Comm. Math. Phys. 359, no. 3, 975-1026 (2018)

work page 2018

[7] [6]

Boccato, C

C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. The excitation spectrum of Bose gases interacting through singular potentials. J. Eur. Math. Soc. 22, no. 7 (2020)

work page 2020

[8] [7]

Boccato, C

C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein. Bog oliubov Theory in the Gross-Pitaevskii limit. Acta Mathematica 222, no. 2, 219-335 (2019)

work page 2019