Propagation of Condensation via Neumann Localization in the Dilute Bose Gas

Propagation of Condensation via Neumann Lo calization in the Dilute Bose Gas Luk as Junge Marc h 31, 2026 Abstract W e pro ve a Neumann lo calization inequality for the Laplacian that includes a sp ectral gap. This result is obtained b y partitioning a cube into o verlapping families of subcub es and analysing the asso ciated pro jection op erators. The resulting operator inequality goes through a discrete Neumann Laplacian on the lattice of b o xes and yields a quantitativ e sp ectral gap estimate. As an application, we consider the dilute Bose gas with Neumann b oundary conditions. Combining the lo calization metho d with recently established free-energy b ounds [4], w e propagate strong condensation estimates from the Gross–Pitaevskii scale to larger b oxes of side length R ∼ a ( ρa 3 ) − 3 4 − η . In tro duction Since the birth of mathematical physics the rigorous deriv ation of Bose–Einstein condensation (BEC) from the microscopic many-bo dy Hamiltonian has remained a cen tral problem. While the ground state energy of the dilute Bose gas is b eing deriv ed to higher and higher precision, establishing condensation directly from first principles — particularly at p ositiv e temp erature and on ph ysically relev an t length scales — is substan tially more delicate. Although the thermo dynamic limit is still not within reach, there ha ve b een significan t con tributions in extending the scale at which we can prov e BEC, see for instance [1, 3, 5, 10]. Recen t adv ances hav e provided precise control of the free energy and established strong condensation estimates in Neumann b o xes whose side lengths are sligh tly larger than the Gross–Pitaevskii scale; see in particular [4] and [8]. These results yield quantitativ e b ounds on the num b er of excited particles, but only on relativ ely short spatial sca les. The purp ose of this pap er is to extend such condensation estimates to longer length scales similar to what was done in the p erio dic setting in [5] based upon metho ds of [2, 6]. Con trary to [5] this is done in the Neumann setting, which is a non trivial v ariation. This technique also leads to a kinetic op erator significan tly simpler than what was used in [6, 7]. In [3] the authors did something similar, how ev er they needed to sacrifice a fraction of the kinetic energy whic h had significant cost to their condensation estimates, this w e a void completely here. The key technical input is a lo calization technique: b y partitioning the domain into o verlapping families of boxes and analysing the asso ciated pro jection operators, we derive a lo w er b ound that uses a discrete Neumann Laplacian with vertices b eing the boxes and edges connecting neighbouring b o xes. Sp ectral analysis of this op erator yields an estimate, which can b e iterated across scales. The result is optimal in the scale parameter. This Neumann localization allows us to propagate the strong condensation estimates obtained on the Gross–Pitaevskii scale [4] to weak condensation estimates on significantly larger b o xes. In particular, w e sho w that condensation persists on scales R ∼ a ( ρa 3 ) − 3 / 4 − η and at temperatures T ∼ ρa . Thereby extending the regime in which BEC can b e rigorously established. This is done in the final section. The Neumann lo calisation W e consider the Neumann Laplacian − ∆ Λ R on a b o x Λ R = [0 , R ] 3 . Its low est eigenv alue is 0 with corre- sp onding eigenv ector the constant function 1 Λ R . Let P denote the pro jection onto the supspace spanned by 1 1 Λ R and Q = 1 − P . F or any subset A ⊂ Λ R , we denote P A the pro jection onto constant functions on A and Q A = 1 A − P A . Our goal is to sub divide Λ R = ⊔ i A i suc h that − ∆ Λ R − Q R 2 ≥ X i ( − ∆ A i − cλ 1 ( A i ) Q A i ) , (1) with λ 1 b eing the first non-zero eigenv alue of − ∆ A i and c > 0 a universal constant. In a quadratic form sense, w e ha ve the op erator inequality − ∆ Λ R ≥ X i − ∆ A i . Th us in order to es tablish eq. (1) it w ould suffice to sho w that Q R 2 ≤ X i cλ 1 ( A i ) Q A i . Ho wev er, this inequality is false in general, since all the op erators Q A i ma y v anish sim ultaneously even when the function is not globally constant.T o ov ercome this issue, we introduce m ultiple ov erlapping partitions of Λ R . The ov erlaps ensure that one can instead prov e an inequality of the form Q R 2 ≤ X k X i cλ 1 ( A i,k ) Q A i,k . (2) In what follo ws, we will restrict our atten tion to the case where the sets A are cub es, as this is the only shape directly applicable to the Bose gas. Since the target inequality is scale-inv arian t, it suffices to establish it for R = 1 Theorem 1 uses tw o partitions in an elegant and optimal w ay . Ho w ever the boundary sets of one of the partitions are not cub es and the theorem is therefore not directly applicable to the Bose gas. In Theorem 3 w e consider many partitions consisting only of cub es. While the underlying idea remains the same, the additional tec hnicalities mak e the argumen t somewhat less transparen t. Theorem 1. F or ℓ − 1 ∈ N , we c onsider the two p artitions [ k ∈{ 0 , 1 ,..,ℓ − 1 − 1 } 3 B ℓ,k = [0 , 1] 3 = [ k ∈{ 0 , 1 ,..,ℓ − 1 − 1 } 3 B shif t ℓ,k . (3) wher e B ℓ,k = 3 Y i =1 [ k i ℓ, ( k i + 1) ℓ ] , B shif t ℓ,k = 3 Y i =1 [( k i + 1 2 1 k i > 0 ) ℓ, ( k i + 1 + 1 2 1 k i <ℓ − 1 − 1 ) ℓ ] . Then for al l ℓ − 1 ∈ N , the fol lowing op er ator ine quality holds on L 2 ([0 , 1] 3 ) : X k Q B ℓ,k + Q B shif t ℓ,k ≥ 1 − cos π ℓ 4 + 1 − cos πℓ Q. (4) Note that the sets B shif t ℓ,k ar e not al l cub es: intervals adjac ent to the b oundary at 1 have length ℓ 2 , while those adjac ent to the b oundary at 0 have length 3 ℓ 2 . Remark 2. The c onstant app e aring in e q. (4) is optimal in the limit ℓ → 0 . Inde e d by defining the function u = 2 ℓ − 1 − 1 X s =0 cos ( π ℓ 2 ( s + 1 2 ))1 s ℓ 2 ≤ x 1 ≤ ( s +1) ℓ 2 , (5) and evaluating e q. (4) on u yields ⟨ u, X k  Q B ℓ,k + Q B shif t ℓ,k  u ⟩ ≥ ( π 2 ℓ 2 8 − O ( ℓ 3 )) ∥ Qu ∥ 2 . (6) Which establishes the optimality of e q. (4) . 2 Pr o of. Since b oth sides of eq. (4) v anish on constant functions, it suffices to v erify the inequalit y on the orthogonal complement of constan t functions. Because the b o xes B ℓ,k are disjoint, up to measure zero, the op erators Q ℓ := X k Q B ℓ,k , Q shif t ℓ := X k Q B shif t ℓ,k (7) are pro jections. W e define P ℓ and P shif t ℓ analogously , naturally Q ℓ = 1 − P ℓ . By Cauc hy-Sc hw artz inequalit y w e find P ℓ Q shif t ℓ P ℓ = Q shif t ℓ − Q ℓ Q shif t ℓ − Q shif t ℓ Q ℓ + Q ℓ Q shif t ℓ Q ℓ ≤ 2( Q ℓ + Q shif t ℓ ) . (8) Th us w e are reduced to studying the op erator L ( ℓ ) = P ℓ Q shif t ℓ P ℓ = P ℓ − P ℓ P shif t ℓ P ℓ . (9) The op erator P ℓ maps L 2 ([0 , 1] 3 ) onto the subspace of functions that are constant on each b ox B ℓ,k . Explicitly , P ℓ ( ψ ) = X k 1 ℓ 3 ˆ B ℓ,k ψ dx ! 1 B ℓ,k This yields a natural isometry b et ween P ℓ L 2 ([0 , 1] 3 ) and ℓ 2 ( { 0 , 1 , .., ℓ − 1 − 1 } 3 ), giv en b y ( Gψ )( k ) = 1 ℓ 3 ˆ B ℓ,k ψ dx, k ∈ { 0 , 1 , .., ℓ − 1 − 1 } 3 . Through this isometry we view L ( ℓ ) as an op erator on ℓ 2 ( { 0 , 1 , .., ℓ − 1 − 1 } 3 ). A direct computation yields ( L ( ℓ ) f )( k ) = X s X s ′ | B ℓ,k ∩ B shif t ℓ,s ′ || B ℓ,s ∩ B shif t ℓ,s ′ | ℓ 3 | B shif t ℓ,s ′ | ( f ( k ) − f ( s )) Ev aluating the ov erlaps explicitly , one finds for k aw ay from the b oundary , ( L ( ℓ ) f )( k ) = 1 16     X | k − s | 1 =1 f ( k ) − f ( s ) + 1 2 X | k − s | ∞ =1 | k − s | 1 =2 f ( k ) − f ( s ) + 1 4 X | k − s | ∞ =1 | k − s | 1 =3 f ( k ) − f ( s )     . (10) One can easily c hec k that the edges connecting to the boundary hav e higher weigh ts. Therefore, as op erators on ℓ 2 ( { 0 , 1 , .., ℓ − 1 − 1 } 3 ), w e find L ( ℓ ) ≥ − ∆ Graph (11) with − ∆ Graph b eing 1 / 16 times the free/Neumann Laplacian on the cubic lattice, with edges of weigh t 1 b et w een nearest neighbours, edges of weigh t 1 2 b et ween ”face diagonal” vertices, and edges of weigh t 1 4 b et ween ”b o dy diagonal” vertices. − ∆ Graph is diagonal in the discrete Neumann basis. More precisely , − ∆ Graph ϕ m = λ m ϕ m , with ϕ m ( k ) = 3 Y i =1 cos( m i π ℓ ( k i + 1 2 )) , m ∈ { 0 , 1 , 2 , .., ℓ − 1 − 1 } 3 λ m = 1 8 X i (1 − cos( π m i ℓ )) + 1 8 X i 0 indep endent of ℓ , such that the fol lowing op er ator ine quality holds on L 2 ([0 , 1] 3 ) : ℓ ⌊ 1 2 ℓ ⌋ X n =0 Q ℓ n ≥ C ℓ 2 Q. (15) Pr o of. Using the same inequality as in eq. (8) w e ha v e for eac h pair ℓ n , ℓ m P ℓ n Q ℓ m P ℓ n ≤ 2( Q ℓ n + Q ℓ m ) , summing o v er n and m yields ⌊ 1 2 ℓ ⌋ X n =0 Q ℓ n ≥ C ℓ X m,n P ℓ m Q ℓ n P ℓ m . Th us, it suffices to pro v e that for eac h m ; L ( ℓ m ) = X n P ℓ m Q ℓ n P ℓ m ≥ C ℓ P ℓ m . (16) Pro ceeding as in eq. (10) w e represent L ( ℓ m ) as the discrete op erator: ( L ( ℓ m ) f )( k ) = X n X s X s ′ | B ℓ m ,k ∩ B ℓ n ,s ′ || B ℓ m ,s ∩ B ℓ n ,s ′ | ℓ 3 m ℓ 3 n ( f ( k ) − f ( s )) . (17) W e b ound L ( ℓ m ) from b elo w by restricting to nearest-neighbour edges, i.e. | k − s | = 1, and simply b ound the diagonal edges b y 0. Without loss of generality assume k 1  = ℓ − 1 m − 1 and consider the weigh t of the edge b et ween k and k + e 1 . X n X s ′ | B ℓ m ,k ∩ B ℓ n ,s ′ || B ℓ m ,k + e 1 ∩ B ℓ n ,s ′ | ℓ 3 m ℓ 3 n ≥ C X n ℓ − 1 n − 1 X s ′ 1 =0 | [ k 1 ℓ m , ( ℓ m + 1) k 1 ] ∩ [ s ′ 1 ℓ n , ( s ′ 1 + 1) ℓ n ] | [( k 1 + 1) ℓ m , ( ℓ m + 2) k 1 ] ∩ [ s ′ 1 ℓ n , ( s ′ 1 + 1) ℓ n ] | ℓ 2 . Using the expression ℓ n = ℓ 1 − nℓ the ab o ve expression can b e b ounded from b elo w by C X n X s ′ 1  k + 1 1 − mℓ − s ′ 1 1 − nℓ  +  s ′ 1 + 1 1 − nℓ − k + 1 1 − mℓ  + . 4 The inner sum ov er s ′ 1 can b e estimated by the following: X s ′ 1 ( ... ) ≥ C d  ( k + 1)(1 − nℓ ) 1 − mℓ , Z  = C d  ( k + 1)( m − n ) ℓ − 1 − m , Z  , (18) where d ( · , Z ) is distance to nearest in teger. Lastly since k 1 +1 ℓ − 1 − m is not an integer, and its distance to an in teger is at least ℓ m ≥ ℓ w e ma y conclude ⌊ 1 2 ℓ ⌋ X n =0 d  ( k + 1)( m − n ) ℓ − 1 − m , Z  ≥ C ℓ − 1 . It follows that L ( ℓ m ) dominates the nearest neighbour discrete Laplacian with edge weigh t C ℓ − 1 . Using the same diagonalization as in eq. (12) yields eq. (16). Application to the dilute Bose gas W e consider the dilute Bose gas describ ed by the Hamiltonian H N = N X i =1 − ∆ i + X i 0. This definition is consisten t with standard notions of Bose–Einstein condensation; see, for instance, [13]. As mentioned in the introduction w e are not able to prov e BEC in the thermo dynamic limit, instead we consider regimes in which N and ρa 3 are coupled b y letting L = af ( ρa 3 ). In [4] the following result was established. Theorem 4. L et v ≥ 0 b e r adial ly de cr e asing and c omp actly supp orte d, then ther e exist C, η > 0 such that for L = a ( ρa 3 ) − 1 2 − η and T ≤ ρa ( ρa 3 ) − η we have − T log T r e − β ( H N − η n + L 2 ) ≥ 4 π ρaN (1 + 128 15 √ π p ρa 3 − C ( ρa 3 ) 1 2 + η ) + L 3 T 5 2 (2 π ) 3 ˆ R 3 log(1 − e − q p 4 + 16 πp 2 ρa T ) dp (23) 5 Remark 5. A t zer o temp er atur e, one c an tr ack the p ar ameter η explicitly in [4]. The optimal value obtaine d ther e is η = 1 32 (24) F rom the v ariational principle eq. (20), we find T r( H N Γ 0 ) + T T r(Γ 0 log Γ 0 ) ≥ − T log T r ( e − β ( H N − η n + L 2 ) ) + T r( η n + L 2 Γ 0 ) Com bining this with Theorem 4 and a corresp onding upp er b ound on the free energy [9], we obtain T r( n + Γ 0 ) ≤ C L 2 N ρa ( ρa 3 ) 1 2 + η ≤ C N ( ρa 3 ) 1 2 − η . (25) This yields a strong condensation estimate on boxes whose side length is sligh tly larger than the Gross–Pitaevskii scale, whic h corresp onds to L = a ( ρa 3 ) − 1 2 . W e no w apply Theorem 3 to extend this strong condensation estimate to a weak condensation estimate on larger length scales. This was similarly done in [5] for the p erio dic setting. Corollary 6. L et v , T , η and C b e given as in The or em 4. Then for R ≥ a ( ρa 3 ) − 1 2 − η we have T r( n + e − β H N ) T r( e − β H N ) ≤ C N ρR 2 a ( ρa 3 ) 1 2 + η . (26) with H N given as in e q. (19) define d on L 2 ([0 , R ] 3 N ) satisfying N = ρR 3 . Remark 7. In much of the Bose gas liter atur e, one fixes the length L = 1 and c onsiders H N ( κ ) = N X i =1 − ∆ i + X i 0, F ( N , L n ) − µN ≥ inf P i m i = N X i F L n ( m i ) − µm i . (28) A t low densit y , Theorem 4 implies that F L n ( m ) is p ositive and essentially quadratic in m . By further sub dividing particles if necessary to main tain low density , w e obtain F L n ( m i ) − µm i ≥ 0 m i ≥ 3 µ 4 π L 3 n where in the ab o ve we use that µ will b e c hosen prop ortionally to ρ . W e can then throw aw a y the positive terms in the sum eq. (28), and insert eq. (23) to find F ( N , L n ) − µN ≥ inf P i m i = N X i,m i ≤ 3 µ 4 π L 3 n 4 π a m 2 i L 3 n (1 + 128 15 √ π r m i L 3 n a 3 − C ( m i L 3 n a 3 ) 1 2 + η ) − µm i + L 3 n T 5 2 (2 π ) 3 ˆ R 3 log(1 − e − r p 4 + 16 πp 2 m i L 3 n a T ) dp. The right-hand side is conv ex in m i , and choosing µ appropriately ensures that the minimum is attained at m i = ρL 3 n . This yields µ ∼ 8 π ρ (1 + O ( p ρa 3 )). The infimum is th us ac hieved with each b o x having exactly ρL 3 n particles and com bining the ab ov e with eq. (27) we find − T log T r e − β ( H N − η n + R 2 ) ≥ 4 π ρaN (1 + 128 15 √ π p ρa 3 − C ( ρa 3 ) 1 2 + η ) + R 3 T 5 2 (2 π ) 3 ˆ R 3 log(1 − e − q p 4 + 16 πp 2 ρa T ) dp. As b efore, combining this low er b ound with the corresp onding upp er b ound [9] and the v ariational principle yields the desired result. A cknow le dgements : This w ork was funded by the Deutsc he F orsc h ungsgemeinschaft (DFG, German Researc h F oundation) – Pro ject-ID 470903074 – TRR 352. The author is grateful for the motiv ation and v aluable discussions with Arnaud T riay and Søren F ournais. References [1] Christian Brenneck e, Morris Bro oks, Cristina Caraci, and Jakob Oldenburg. “A Short Pro of of Bose–Einstein Condensation in the Gross–Pitaevskii Regime and Beyond”. In: Annales Henri Poinc ar ´ e (2025). doi : 10.1007/s00023- 024- 01465- 8 . url : https://doi.org/10.1007/s00023- 024- 01465- 8 . [2] Birger Brietzke and Jan Solov ej. “The Second-Order Correction to the Ground State Energy of the Dilute Bose Gas”. In: Ann. Inst. Henri Poinc ar´ e 21 (Jan. 2020). doi : 10.1007/s00023- 019- 00875- 3 . [3] Jac ky J. Chong, Hao Liang, and Phan Th` anh Nam. Kinetic lo c alization via Poinc ar´ e-typ e ine qualities and applic ations to the c ondensation of Bose gases . 2025. arXiv: 2510.20493 [math-ph] . url : https: //arxiv.org/abs/2510.20493 . 7 REFERENCES REFERENCES [4] S. F ournais, L. Junge, T. Girardot, L. Morin, M. Olivieri, and A. T riay. The fr e e ener gy of dilute Bose gases at low temp er atur es inter acting via str ong p otentials . 2024. arXiv: 2408.14222 [math-ph] . url : https://arxiv.org/abs/2408.14222 . [5] Søren F ournais. “Length scales for BEC in the dilute Bose gas”. In: Partial Differ ential Equations, Sp e c- tr al The ory, and Mathematic al Physics (2020). url : https://api.semanticscholar.org/CorpusID: 260436492 . [6] Søren F ournais and Jan Philip Solo v ej. “The energy of dilute Bose gases”. In: A nn. of Math. 192.3 (2020), pp. 893–976. doi : 10.4007/annals.2020.192.3.5 . url : https://doi.org/10.4007/annals. 2020.192.3.5 . [7] Søren F ournais and Jan Philip Solov ej. “The energy of dilute Bose gases I I: general case”. In: Inven- tiones mathematic ae (2022). doi : 10. 1007/s00222- 022- 01175- 0 . url : https:/ /doi.org/10. 1007/ s00222- 022- 01175- 0 . [8] Florian Hab erberger, Christian Hainzl, Phan Th` anh Nam, Rob ert Seiringer, and Arnaud T ria y. The fr e e ener gy of dilute Bose gases at low temp er atur es . 2024. arXiv: 2304.02405 [math-ph] . url : https: //arxiv.org/abs/2304.02405 . [9] Florian Hab erb erger, Christian Hainzl, Benjamin Schlein, and Arnaud T riay. Upp er Bound for the F r e e Ener gy of Dilute Bose Gases at L ow T emp er atur e . 2024. arXiv: 2405 . 03378 [math-ph] . url : https://arxiv.org/abs/2405.03378 . [10] Elliott H. Lieb and Robert Seiringer. “Proof of Bose-Einstein Condensation for Dilute T rapp ed Gases”. In: Phys. R ev. L ett. 88 (17 Apr. 2002), p. 170409. doi : 10 . 1103 / PhysRevLett . 88 . 170409 . url : https://link.aps.org/doi/10.1103/PhysRevLett.88.170409 . [11] Elliott H. Lieb, Rob ert Seiringer, Jan Philip Solov ej, and Jakob Yngv ason. The Mathematics of the Bose Gas and its Condensation . 2005. url : https://doi.org/10.1007/b137508 . [12] Elliott H. Lieb and Jak ob Yngv ason. “Ground State Energy of the Low Density Bose Gas”. In: Phys. R ev. L ett. 80.12 (Mar. 1998), pp. 2504–2507. doi : 10.1103/physrevlett.80.2504 . [13] Oliv er Penrose and Lars Onsager. “Bose-Einstein Condensation and Liquid Helium”. In: Phys. R ev. 104 (3 Nov. 1956), pp. 576–584. doi : 10.1103/PhysRev.104.576 . url : https://link.aps.org/doi/ 10.1103/PhysRev.104.576 . Deparmen t of mathematics, Ludwig-Maximilians-Universit¨ at-M ¨ unc hen, Germany Email: luk as.junge@math.lmu.de 8