The Average Field Approximation for Almost Bosonix Extended Anyons
aa r X i v : . [ m a t h - ph ] S e p THE AVERAGE FIELD APPROXIMATION FOR ALMOST BOSONICEXTENDED ANYONS
DOUGLAS LUNDHOLM AND NICOLAS ROUGERIE
Abstract.
Anyons are 2D or 1D quantum particles with intermediate statistics, inter-polating between bosons and fermions. We study the ground state of a large number N of 2D anyons, in a scaling limit where the statistics parameter α is proportional to N − when N → ∞ . This means that the statistics is seen as a “perturbation from the bosonicend”. We model this situation in the magnetic gauge picture by bosons interacting throughlong-range magnetic potentials. We assume that these effective statistical gauge potentialsare generated by magnetic charges carried by each particle, smeared over discs of radius R (extended anyons). Our method allows to take R → N → ∞ . In this limit we rigorously justify the so-called “average field approxima-tion”: the particles behave like independent, identically distributed bosons interacting viaa self-consistent magnetic field. Contents
1. Introduction 11.1. The model 21.2. Average field approximation 41.3. Average field versus mean field 51.4. Main results 72. The extended anyon Hamiltonian 92.1. Operator bounds for the interaction terms 92.2. A priori bound for the ground state 153. Mean-field limit 163.1. Preliminaries 163.2. Truncated Hamiltonian 183.3. Energy bounds 223.4. Convergence of states 23Appendix A. Properties of the average-field functional 25References 301.
Introduction
In lower dimensions there are possibilities for quantum statistics different from bosonsand fermions, so called intermediate or fractional statistics. Due to the prospect that suchparticles, termed anyons (as in any thing in between bosons and fermions), could arise as
Date : September, 2015. effective quasiparticles in many-body quantum systems confined to lower dimensions, therehas been a great interest over the last three decades in figuring out the behavior of suchstatistics (see [15, 19, 20, 23, 39, 42, 53] for extensive reviews). In one dimension, one canview the Lieb-Liniger model [28] as providing an example of effective interpolating statistics.Although initially regarded as purely hypothetical while at the same time offering substan-tial analytical insight thanks to its exact solvability, this system has now been realizedconcretely in the laboratory [21, 43]. Much less is known concerning fractional statistics inthe two-dimensional setting, conjectured [1] to be relevant for the fractional quantum Halleffect (see [16, 22] for review), and it is indeed a very challenging theoretical question tofigure out even the ground state properties of an ideal 2D many-anyon gas, parameterizedby a single statistics phase e iπα , or a periodic real parameter α (with α = 0 correspondingto bosons and α = 1 to fermions). On the rigorous analytical side, some recent progress inthis direction has been achieved in [34, 35, 36] where a better understanding of the groundstate energy was obtained in the case that α is an odd numerator fraction. Numerous ap-proximative descriptions have also been proposed over the years, such as e.g. in [6] wherethe problem was approached from both the bosonic and the fermionic ends, with a harmonictrapping potential. Here, equipped with new methods in many-body spectral theory, wewill re-visit this question from the perspective of a perturbation around bosons, i.e. in aregime where α is small.1.1. The model.
One may formally think of 2D anyons with statistics parameter α asbeing described by an N -body wave function of the formΨ( x , . . . , x N ) = Y j
The few-anyons problem can be studied within per-turbation theory, yielding satisfactory information on the ground state and low-lying exci-tation spectrum [48, 9, 41, 10]. For many anyons however, it is hard to obtain results thisway. A possible approximation to obtain a more tractable model when N is large consistsin seeing the potential (1.2) or (1.5) as being independent of the precise positions x j andinstead generated by the mean distribution of the particles (whence the name, average fieldapproximation [53]) A [ ρ ] := ∇ ⊥ w ∗ ρ, A R [ ρ ] := ∇ ⊥ w R ∗ ρ, (1.9)where ρ is the one-body density (normalized in L ( R )) of a given bosonic wave function Ψ ρ ( x ) = Z R N − | Ψ( x, x , . . . , x N ) | dx . . . dx N , By the boundedness of ∇ w R and using Cauchy-Schwarz, all terms are infinitesimally form-bounded interms of H N ( α = 0) and hence H RN is a uniquely defined self-adjoint operator by the KLMN theorem [45,Theorem X.17]. We shall assume V is such that a form core is given by C ∞ c ( R ). VERAGE FIELD APPROXIMATION 5 say the ground-state wave function. One then obtains from (1.6) the approximate N -bodyHamiltonian H af N [ ρ ] := N X j =1 (cid:16)(cid:0) p j + N α A R [ ρ ] (cid:1) + V ( x j ) (cid:17) . If one considers ρ as fixed, the ground state of this non-interacting magnetic Hamiltonianacting on L ( R N ) is a pure Bose condensateΨ N = u ⊗ N , where u ∈ L ( R ) should minimize h u (cid:12)(cid:12) ( p + N α A R [ ρ ]) + V (cid:12)(cid:12) u i = N − D Ψ N , H af N [ ρ ]Ψ N E . For consistency, one should then impose that | u | = ρ, which leads to a non-linear minimization problem. One thus looks for the minimum E af andminimizer u af of the following average-field energy functional (recall the notation β ∼ N α ) E af R [ u ] := Z R (cid:16)(cid:12)(cid:12)(cid:0) ∇ + iβ A R [ | u | ] (cid:1) u (cid:12)(cid:12) + V | u | (cid:17) (1.10)under the unit mass constraint Z R | u | = 1 . Note that, for this problem to be independent of N it is pretty natural to — in line with (1.3)— assume that β ∼ N α is fixed. It is not difficult to see that if
N α → E af R =0 , which defines a strictly two-dimensional modelof particles with a self-generated magnetic field B ( x ) = curl β A [ ρ ]( x ) = 2 πβρ ( x ) withoutpropagating degrees of freedom and to which one could further consider adding an externalmagnetic field, is also of relevance for various Chern-Simons formulations of anyonic theories(see, e.g., [55, 54, 19, 3]).1.3. Average field versus mean field.
In principle, the average field approximation doesnot require that the true ground state of H RN be Bose-condensed. In fact, the most commonapplication of it has been in perturbing around fermions α = 1 [13, 19, 50, 51, 52] (this haseven been argued to be preferable [4, 53]), and usually one even restricts to the homogeneoussetting with ρ a constant. However, the case of fixed β ∼ N α which is natural for the studyof (1.10), places the limit N → ∞ of the original many-body problem in a mean-field-likeregime for bosons. Indeed, observe that in (1.8), the two-body terms in the second lineand the three-body term in the third line weigh a total O ( N ) in the energy in this regime,comparable to the one-body term in the first line. The two-body term in the fourth line isof much smaller order, O (1) roughly, which is fortunate because of its singularity. Actually,if one takes bluntly R = 0, the potential |∇ w | appearing in this term is not locallyintegrable, and hence an ansatz Ψ N = u ⊗ N would lead to an infinite energy. For extendedanyons, R >
D. LUNDHOLM AND N. ROUGERIE • The effective interaction is peculiar: it comprises a three-body term, and a two-bodyterm which mixes position and momentum variables. • The limit problem (1.10) comprises an effective self-consistent magnetic field. Aterm in the form of a self-consistent electric field is more usual. • One should deal with the limit R → N → ∞ , which isreminiscent of the NLS and GP limits for trapped Bose gases [32, 31, 30, 26, 40].In order to make the analogy more transparent, we rewrite, for any normalized N -bodybosonic wave function Ψ N ∈ L ( R N ) N − (cid:10) Ψ N (cid:12)(cid:12) H RN (cid:12)(cid:12) Ψ N (cid:11) = Tr h ( p + V ) γ (1) N i + β Tr h(cid:16) p · ∇ ⊥ w R ( x − x ) + ∇ ⊥ w R ( x − x ) · p (cid:17) γ (2) N i + β N − N − h(cid:16) ∇ ⊥ w R ( x − x ) · ∇ ⊥ w R ( x − x ) (cid:17) γ (3) N i + β N − h |∇ w R ( x − x ) | γ (2) N i , (1.11)where γ ( k ) N := Tr k +1 → N [ | Ψ N ih Ψ N | ]is the k -body density matrix of the state | Ψ N ih Ψ N | , normalized to have trace 1. Thenotation here means that we trace out the last N − k variables from the integral kernel of | Ψ N ih Ψ N | .Since all terms at least at first sight weigh O (1) or less, the folklore suggests to use anansatz Ψ N = u ⊗ N γ ( k ) N = | u ⊗ k ih u ⊗ k | . (1.12)Inserting this in the energy, dropping the last term, which is of order N − at least forfixed R , we obtain to leading order N − (cid:10) Ψ N (cid:12)(cid:12) H RN (cid:12)(cid:12) Ψ N (cid:11) ≈ E af R [ u ] . (1.13)Indeed, on the one hand,Tr h(cid:16) p · ∇ ⊥ w R ( x − x ) + ∇ ⊥ w R ( x − x ) · p (cid:17) | u ⊗ ih u ⊗ | i = i Z Z R × R ∇ u ( x ) u ( y ) · ∇ ⊥ w R ( x − y ) u ( x ) u ( y ) dxdy − i Z Z R × R u ( x ) u ( y ) ∇ ⊥ w R ( x − y ) · ∇ u ( x ) u ( y ) dxdy = 2 Z R A R [ | u | ] · J [ u ] , (1.14)using the definition (1.9) and denoting J [ u ] the current J [ u ] := i u ∇ u − u ∇ u ) . (1.15) VERAGE FIELD APPROXIMATION 7
Note that this is really a phase current density: J [ u ] = ρ ∇ ϕ if u = √ ρe iϕ . On the other handTr h(cid:16) ∇ ⊥ w R ( x − x ) · ∇ ⊥ w R ( x − x ) (cid:17) γ (3) N i = Z Z Z R × R × R | u ( x ) | | u ( y ) | | u ( z ) | ∇ ⊥ w R ( x − y ) · ∇ ⊥ w R ( x − z ) dxdydz = Z R | u | (cid:12)(cid:12) A R [ | u | ] (cid:12)(cid:12) , (1.16)and it suffices to combine these identities in (1.11) (and approximate ( N − / ( N − ∼ Main results.
We may now state our main theorem, justifying the average field ap-proximation in the almost-bosonic limit at the level of the ground state. For technicalreasons we assume that the one-body potential is confining V ( x ) ≥ c | x | s − C, s > , (1.17)and that the size R of the extended anyons does not go to zero too fast in the limit N → ∞ .The rate we may handle depends on s . These assumptions are probably too restrictive froma physical point of view but our method of proof does not allow to relax them at present.Here and in the sequel, E af denotes the average-field functional (1.10) for R = 0, and E af its infimum under a unit mass constraint. Although we do not state it explicitly, we couldalso keep R fixed when N → ∞ and obtain the limit functional with finite R . The caseof anyons in a bounded domain is also covered by our approach (modulo the discussion ofboundary conditions) and the results in this case can be obtained by formally setting s = ∞ in the following. Theorem 1.1 ( Validity of the average field approximation ) . We consider N extended anyons of radius R ∼ N − η in an external potential V satisfy-ing (1.17) . We assume the relation < η < η ( s ) := 14 (cid:18) s (cid:19) − , (1.18) and that the statistics parameter scales as α = β/ ( N − for fixed β ∈ R . Then, in the limit N → ∞ we have for the ground-state energy E R ( N ) N → E af . (1.19) Moreover, if Ψ N is a sequence of ground states for H RN , with associated reduced densitymatrices γ ( k ) N , then modulo restricting to a subsequence we have γ ( k ) N → Z M af | u ⊗ k ih u ⊗ k | dµ ( u ) (1.20) D. LUNDHOLM AND N. ROUGERIE strongly in the trace-class when N → ∞ , where µ is a Borel probability measure supportedon the set of minimizers of E af , M af := { u ∈ L ( R ) : k u k L = 1 , E af [ u ] = E af } . Remark . As we said before, taking R not too small in the limit N → ∞ is a requirement in ourmethod of proof. It is likely that localization arguments could allow to take an s -independent η < η ( ∞ ) = 1 /
4, corresponding to the rate we obtain for anyons in a bounded domain. Toobtain an even better rate would require important new ideas.It is in fact not clear whether some lower bound on R is a necessary condition for theaverage field description to be correct. For very small or zero R , it is still conceivable thata description in terms of a functional of the form of (1.10) is correct in the limit. Indeed,our above restriction stems from the method used to bound the ground-state energy frombelow, while for an upper bound the conditions on R (and even the finiteness of β ) can berelaxed significantly. A further possibility would be to take short-range correlations intoaccount via Jastrow factors, as in the GP limit for the usual Bose gas [32, 31, 30, 26, 40].It would in any case be desirable to be able to take R ≪ N − / , the typical interparticledistance in our setting, because one could then argue that smearing the magnetic chargeshas very little effect. Even in the formal case s = ∞ , Theorem 1.1 requires R ≫ N − / ≫ N − / , which is rather stringent. We nevertheless obtain the functional for point-like anyonsin the limit.It is sometimes argued in the literature [38, 50, 7] that for anyons arising as quasi-particlesin condensed matter physics, the magnetic charges should be smeared over some length scale R . The relevant relation between R and N then depends on the context. ✸ The rest of the paper contains the proof of Theorem 1.1. We start by collecting inSection 2 some operator bounds on the different terms of the N -body functional. This isrequired in order to have a correct control of the terms as a function of the kinetic energyin the limit R →
0. For these estimates to be of use in the large- N limit we need an apriori bound on the kinetic energy of ground states of the N -body problem, also derived inSection 2. We deal with the mean-field limit in Section 3, using the method of [26]. Someimportant adaptations are required to deal with the anyonic Hamiltonian, and we focus onthese. The goal here is to justify (with quantitative error bounds) the sensibility of theansatz Ψ N = u ⊗ N when N becomes large, thus obtaining E af R as an approximation of theground state energy per particle. The basic properties of the average-field functional (1.10)are worked out in Appendix A. In particular we study the limit R → E af as an approximation of the many-body ground state energy per particle. Acknowledgments.
We thank Michele Correggi for discussions. Part of this work hasbeen carried out during visits at the
Institut Henri Poincar´e (Paris) and the
Institut Mittag-Leffler (Stockholm). D.L. would also like to thank LPMMC Grenoble for kind hospitality.We acknowledge financial support from the
French ANR (Project Mathosaq ANR-13-JS01-0005-01), as well as the grant KAW 2010.0063 from the
Knut and Alice Wallenberg Foun-dation and the
Swedish Research Council grant no. 2013-4734.
VERAGE FIELD APPROXIMATION 9 The extended anyon Hamiltonian
In this section we give some bounds allowing to properly define and control the Hamil-tonian (1.6). As previously mentioned, for extended anyons, it is possible to expand theHamiltonian as in (1.8) and estimate it term by term. By the boundedness of the interac-tion it follows that H RN is defined uniquely as a self-adjoint operator on L ( R N ) with thesame form domain as the non-interacting bosonic Hamiltonian N X j =1 ( p j + V ( x j )) . However, in order to eventually take the limit R → R . These will be used to deal with the mean-field limit in Section 3.In the following we introduce a fixed reference length scale R >
0, and always assume R ≪ R . Future constants, generically denoted by C , may implicitly depend on R .2.1. Operator bounds for the interaction terms.
We start with some estimates on thedifferent terms in (1.11), exploiting the regularizing effect of taking
R >
0. The followingis standard:
Lemma 2.1 ( The smeared Coulomb potential ) . Let w R be defined as in (1.4) . There is a constant C > depending only on R such that sup B (0 ,R ) | w R | ≤ C + | log R | , sup R |∇ w R | ≤ CR , sup B (0 ,R ) c |∇ w R | ≤ C. (2.1) Moreover, for any < p < ∞ , k∇ w R k L p ( R ) ≤ C p R /p − . (2.2) Proof.
A simple application of Newton’s theorem [29, Theorem 9.7] yields w R ( x ) = ( log | x | if | x | ≥ R log R + (cid:16) | x | R − (cid:17) if 0 ≤ | x | ≤ R, ∇ w R ( x ) = ( x/ | x | if | x | ≥ Rx/R if 0 ≤ | x | ≤ R, (2.3)and (2.1) clearly follows. For (2.2) we compute k∇ w R k pL p ( R ) = 2 π Z R r p R p rdr + 2 π Z ∞ R r − p rdr ≤ C pp R − p , where C p > p > (cid:3) We first estimate the most singular term of the Hamiltonian, corresponding to the fourthline of (1.8). Since it comes with a relative weight O ( N − ) in the total energy, the followingbound will be enough to discard it from leading order considerations. Lemma 2.2 ( Singular two-body term ) . We have that, as operators on L ( R ) or L ( R ) , |∇ w R ( x − y ) | ≤ C ε R − ε (cid:0) p x + 1 (cid:1) (2.4) for any ε > . Proof.
We start with a well-known simple application of H¨older’s and Sobolev’s inequalities:for any W : R R and f ∈ C ∞ c ( R ) h f | W ( x − y ) | f i = Z Z R × R f ( x, y ) W ( x − y ) f ( x, y ) dxdy ≤ k W k L p Z R (cid:18)Z R | f ( x, y ) | q dx (cid:19) /q dy ≤ C k W k L p Z Z R × R (cid:0) |∇ x f ( x, y ) | + | f ( x, y ) | (cid:1) dxdy = C k W k L p h f | ( − ∆ x + 1) ⊗ | f i (2.5)where we may take any p > q = pp − ∈ (1 , + ∞ ), and we use that in R , for any 1 ≤ q < ∞k g k L q ≤ C q (cid:16) k∇ g k L + k g k L (cid:17) , see, e.g., [29, Theorem 8.5 ii]. Next we may use (2.2) with W = |∇ w R | and p = 1 + ε ′ toconclude k W k L p = k∇ w R k L p ≤ C p R /p − ≤ C ε R − ε , with a constant C ε > ε > (cid:3) We next deal with the two-body term mixing position and momentum, second lineof (1.8). This is somehow the most difficult term to handle, and it is crucial to observethat it acts on the current and not on the full momentum. We shall use three differentbounds. In the following lemma, (2.6) has a worst R -dependence but it behaves better forlarge momenta than (2.7) and (2.8), a fact that will be useful when projecting the problemonto finite dimensional spaces in the next section. Estimate (2.8) might seem a bit betterthan (2.7), but we will actually need a bound on the absolute value in the sequel, which isnot provided by (2.8). Lemma 2.3 ( Mixed two-body term ) . For
R < R small enough we have that, as operators on L ( R ) , (cid:12)(cid:12)(cid:12) p x · ∇ ⊥ w R ( x − y ) + ∇ ⊥ w R ( x − y ) · p x (cid:12)(cid:12)(cid:12) ≤ CR − | p x | , (2.6) (cid:12)(cid:12)(cid:12) p x · ∇ ⊥ w R ( x − y ) + ∇ ⊥ w R ( x − y ) · p x (cid:12)(cid:12)(cid:12) ≤ C ε R − ε ( p x + 1) , for all ε > , (2.7) and ± (cid:16) p x · ∇ ⊥ w R ( x − y ) + ∇ ⊥ w R ( x − y ) · p x (cid:17) ≤ C (1 + | log R | ) ( p x + 1) . (2.8) Proof.
The bounds (2.6) and (2.7) are based on the same basic computation.
Proof of (2.6). First note that p x · ∇ ⊥ w R ( x − y ) = ∇ ⊥ w R ( x − y ) · p x (2.9)because ∇ x · ∇ ⊥ w R ( x − y ) = 0. We can then square the expression we want to estimate,obtaining (cid:16) p x · ∇ ⊥ w R ( x − y ) + ∇ ⊥ w R ( x − y ) · p x (cid:17) = 4 p x · ∇ ⊥ w R ( x − y ) ∇ ⊥ w R ( x − y ) · p x . VERAGE FIELD APPROXIMATION 11
Consequently, for any f = f ( x, y ) ∈ C ∞ c ( R ), (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) f (cid:12)(cid:12) (cid:16) p x · ∇ ⊥ w R ( x − y ) + ∇ ⊥ w R ( x − y ) · p x (cid:17) (cid:12)(cid:12) f (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) = 4 (cid:12)(cid:12)(cid:12)(cid:12)Z Z R × R (cid:16) ∇ x ¯ f ( x, y ) · ∇ ⊥ w R ( x − y ) (cid:17) (cid:16) ∇ x f ( x, y ) · ∇ ⊥ w R ( x − y ) (cid:17) dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Z R × R |∇ x f ( x, y ) | (cid:12)(cid:12)(cid:12) ∇ ⊥ w R ( x − y ) (cid:12)(cid:12)(cid:12) dxdy. Inserting (2.1) we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) f (cid:12)(cid:12) (cid:16) p x · ∇ ⊥ w R ( x − y ) + ∇ ⊥ w R ( x − y ) · p x (cid:17) (cid:12)(cid:12) f (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CR Z Z R × R |∇ x f | dxdy and thus (cid:16) p x · ∇ ⊥ w R ( x − y ) + ∇ ⊥ w R ( x − y ) · p x (cid:17) ≤ CR p x . We deduce (2.6) because the square root is operator monotone (see, e.g., [2, Chapter 5]).
Proof of (2.7). We proceed in the same way but use Lemma 2.2 instead of just the roughbound (2.1) (we denote x = ( x , x ) ∈ R ): (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) f (cid:12)(cid:12) (cid:16) p x · ∇ ⊥ w R ( x − y ) + ∇ ⊥ w R ( x − y ) · p x (cid:17) (cid:12)(cid:12) f (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Z R × R |∇ x f ( x, y ) | (cid:12)(cid:12)(cid:12) ∇ ⊥ w R ( x − y ) (cid:12)(cid:12)(cid:12) dxdy = (cid:28) ∂ x f, (cid:12)(cid:12)(cid:12) ∇ ⊥ w R ( x − y ) (cid:12)(cid:12)(cid:12) ∂ x f (cid:29) L ( R ) + (cid:28) ∂ x f, (cid:12)(cid:12)(cid:12) ∇ ⊥ w R ( x − y ) (cid:12)(cid:12)(cid:12) ∂ x f (cid:29) L ( R ) ≤ C ε R ε ( h ∂ x f, ( − ∆ x + 1) ∂ x f i + h ∂ x f, ( − ∆ x + 1) ∂ x f i ) ≤ C ε R ε D f, ( − ∆ x + 1) f E . Thus (cid:16) p x · ∇ ⊥ w R ( x − y ) + ∇ ⊥ w R ( x − y ) · p x (cid:17) ≤ C ε R ε ( p x + 1) for any ε >
0, and the desired bound again follows by taking the square root.
Proof of (2.8). The idea is here a bit different. We pick f ∈ C ∞ c ( R ; C ) and compute asin (1.14) D f (cid:12)(cid:12) p x · ∇ ⊥ w R ( x − y ) + ∇ ⊥ w R ( x − y ) · p x (cid:12)(cid:12) f E = 2 Z Z R × R ∇ ⊥ w R ( x − y ) · J x [ f ] dxdy with J x [ f ] = i (cid:0) f ∇ x f − f ∇ x f (cid:1) . We then split this according to a partition of unity χ + η = 1 where χ ≡ B (0 , R ) and η ≡ B (0 , R ): D f (cid:12)(cid:12) p x · ∇ ⊥ w R ( x − y ) + ∇ ⊥ w R ( x − y ) · p x (cid:12)(cid:12) f E = 2 Z Z R × R ∇ ⊥ ( χ ( x − y ) w R ( x − y )) · J x [ f ] dxdy + 2 Z Z R × R ∇ ⊥ ( η ( x − y ) w R ( x − y )) · J x [ f ] dxdy. To control the χ term we use Stokes’ formula and deduce2 Z Z R × R ∇ ⊥ ( χ ( x − y ) w R ( x − y )) · J x [ f ] dxdy = − Z Z R × R χ ( x − y ) w R ( x − y )curl x J x [ f ] dxdy. It is easy to see that | curl x J x [ f ] | ≤ |∇ x f | pointwise, see e.g. [11, Lemma 3.4]. We thus obtain ± Z Z R × R ∇ ⊥ ( χ ( x − y ) w R ( x − y )) · J x [ f ] dxdy ≤ Z Z R × R | χ ( x − y ) || w R ( x − y ) | |∇ x f ( x, y ) | dxdy ≤ C (1 + | log R | ) Z Z R × R |∇ x f ( x, y ) | dxdy in view of (2.1). For the η term we note that | J x [ f ] | ≤ | f ||∇ x f | . Thus ± Z Z R × R ∇ ⊥ ( η ( x − y ) w R ( x − y )) · J x [ f ] dxdy ≤ Z Z R × R (cid:12)(cid:12)(cid:12) ∇ ⊥ ( η ( x − y ) w R ( x − y )) (cid:12)(cid:12)(cid:12) | f ( x, y ) | dxdy + Z Z R × R |∇ x f ( x, y ) | dxdy ≤ C Z Z R × R ¯ f ( − ∆ x + 1) f dxdy. For the first term we used that (2.3) implies (cid:12)(cid:12)(cid:12) ∇ ⊥ ( ηw R ) (cid:12)(cid:12)(cid:12) ≤ |∇ η || w R | + | η ||∇ w R | ≤ C because η ≡ B (0 , R ) and η ≡ B (0 , R ). Gathering these estimates weobtain the desired operator bound. (cid:3) The three-body term (third line of (1.8)) is actually a pretty regular potential term, asshown in the following:
Lemma 2.4 ( Three-body term ) . We have that, as operators on L ( R ) , ≤ ∇ ⊥ w R ( x − y ) · ∇ ⊥ w R ( x − z ) ≤ C ( p x + 1) . (2.10)The essential ingredient of the proof is the following three-particle Hardy inequality of [18,Lemma 3.6] (see also [33] for relevant methods and generalizations): VERAGE FIELD APPROXIMATION 13
Lemma 2.5 ( Three-body Hardy inequality ) . Let d ≥ and u : R d → C . Let R ( x, y, z ) be the circumradius of the triangle with vertices x, y, z ∈ R d , and ρ ( x, y, z ) := p | x − y | + | y − z | + | z − x | . Then R − ≤ ρ − pointwise,and d − Z R d | u ( x, y, z ) | ρ ( x, y, z ) dxdydz ≤ Z R d (cid:16) |∇ x u | + |∇ y u | + |∇ z u | (cid:17) dxdydz. (2.11) Proof of Lemma 2.4.
Since we consider the operator as acting on symmetric wave functionsit is equivalent to estimate X cyclic in x, y, z ∇ ⊥ w R ( x − y ) · ∇ ⊥ w R ( x − z ) . (2.12)In general, let x, y, z ∈ R d denote the vertices of a triangle and | x | R := max {| x | , R } aregularized distance. Then we claim the following geometric fact:0 ≤ X cyclic in x, y, z x − y | x − y | R · x − z | x − z | R ≤ Cρ ( x, y, z ) . (2.13)Recalling (2.3) this gives a control on the expression we are interested in. Equivalently, weshall prove that0 ≤ | y − z | R ( x − y ) · ( x − z ) + | z − x | R ( y − z ) · ( y − x ) + | x − y | R ( z − x ) · ( z − y ) ≤ C | x − y | R | y − z | R | z − x | R | x − y | + | y − z | + | z − x | , (2.14)for some constant C > R .Let us consider each of the different geometric configurations that may occur. In thecase that all edge lengths of the triangle are greater than R , the cyclic expression that wewish to estimate in (2.13) reduces to R ( x, y, z ) − (see [18, Lemma 3.2]), which is clearlynon-negative and bounded by ρ − from Lemma 2.5. On the other hand, if all edge lengthsare smaller than R then the expression equals R ρ (cf. [18, Lemma 3.4]), for which wehave 0 ≤ R ρ ≤ ρ − since ρ ≤ R . If two of the edges are short and one long, say | x − y | , | y − z | ≤ R and | z − x | ≥ R , then the expression to be estimated in (2.14) reads R ( x − y ) · ( x − z ) + | z − x | ( y − z ) · ( y − x ) + R ( z − x ) · ( z − y ) | {z } − ( z − y ) · ( x − z ) = R (( x − y ) − ( z − y ))) · ( x − z ) + | x − z | ( y − z ) · ( y − x )= | x − z | (cid:16) R + ( y − z ) · ( y − x ) (cid:17) ≥ | x − z | ( R − R ) ≥ . We furthermore have the upper bound | x − z | (cid:16) R + ( y − z ) · ( y − x ) (cid:17) ≤ R | x − z | , while the r.h.s. of (2.14) is larger than R | x − z | R + | x − z | ≥ R | x − z | , using that | x − z | ≤ | x − y | + | y − z | ≤ R . This leaves the case that only one edge is short, say | x − y | ≤ R , and the others long, | y − z | , | z − x | ≥ R . We thus consider the expression in (2.14) | y − z | ( x − y ) · ( x − z ) + | z − x | ( y − z ) · ( y − x ) + R ( z − x ) · ( z − y ) . (2.15)We will here use methods from [33], namely the geometric (Clifford) algebra G ( R d ) over R d (see [37] for a general introduction). In the case d = 2 or d = 3 one can think of this as thereal algebra generated by the Pauli matrices σ j , with scalar projection h A i := Tr A andthe embedding of scalars (0-vectors) 1 ֒ → and of 1-vectors R d ∋ x ֒ → P dj =1 x j σ j ∈ G ( R d ),and with the product of two 1-vectors xy = x · y + x ∧ y decomposing into a tracefulsymmetric scalar part and a traceless antisymmetric bivector part. We have then, usingtracelessness of the bivector parts of such products and the linearity and cyclicity of thetrace, | y − z | ( x − y ) · ( x − z ) + | z − x | ( y − z ) · ( y − x )= (cid:10) ( y − z ) ( x − y )( x − z ) + ( z − x ) ( y − z )( y − x ) (cid:11) = h ( y − z )( x − y )( x − z )( y − z ) + ( y − x )( z − x )( z − x )( y − z ) i = h ( y − z )( x − y )( z − x )( z − y ) + ( x − y )( z − x )( z − x )( z − y ) i = D(cid:16) ( y − z )( x − y ) + ( z − x )( x − y ) + 2( x − y ) ∧ ( z − x ) (cid:17) ( z − x )( z − y ) E = h ( y − x )( x − y )( z − x )( z − y ) i + 2 (cid:10) ( x − y ) ∧ ( z − x ) ( z − x )( z − y ) (cid:11) = − (cid:10) ( x − y ) ( z − x )( z − y ) (cid:11) + 2 (cid:10) ( x − y ) | {z } z − y − ( z − x ) ∧ ( z − x ) ( z − x ) ∧ ( z − y ) (cid:11) = −| x − y | ( z − x ) · ( z − y ) + 2 h B † B i , with B := ( z − x ) ∧ ( z − y ) and its Hermite conjugate B † = ( z − y ) ∧ ( z − x ). In the fourth andfifth steps we used xy = yx + 2 x ∧ y for the second term and then ( y − z ) + ( z − x ) = y − x ,while for the final steps we again used the properties of the trace and that B † B = | B | isscalar. Thus, the expression (2.15) we wish to estimate equals (cid:0) R − | x − y | (cid:1) ( z − x ) · ( z − y ) + 2 | B | ≥ , where for the lower bound we also used that ( z − x ) · ( z − y ) ≥ x − y is the shortestedge. For an upper bound we can use permutation invariance (cf. [33, Proposition 15]) of | B | = | ( x − y ) ∧ ( x − z ) | ≤ | x − y || x − z | , and for example that | y − z | ≤ R + | x − z | ≤ | x − z | . Hence (cid:0) R − | x − y | (cid:1) ( z − x ) · ( z − y ) + 2 | B | ≤ R | x − z | , while for the r.h.s. of (2.14), with analogously | x − z | ≤ | y − z | , R | y − z | | z − x | R + | y − z | + | z − x | ≥ R | y − z | | z − x | | y − z | = 16 R | x − z | . We also remark that the non-negativity of (2.13) is in general false if | · | R is replaced by anarbitrary radial function, as can be checked when taking e.g. | x | R = e | x | / .Finally, the estimate (2.10) follows simply by applying Lemma 2.5 with d = 2 to (2.13)and using the symmetry of functions in L ( R ). (cid:3) VERAGE FIELD APPROXIMATION 15
A priori bound for the ground state.
For the estimates of the previous subsectionto apply efficiently, we need an a priori bound on ground states (or approximate groundstates) of the N -body Hamiltonian (1.6), provided in the following: Proposition 2.6 ( A priori bound for many-body ground states ) . Let Ψ N ∈ L ( R N ) be a (sequence of ) approximate ground states for H RN , that is, h Ψ N , H RN Ψ N i ≤ E R ( N )(1 + o (1)) when N → ∞ . Denote by γ (1) N the associated sequence of one-body density matrices. In the regime (1.3) ,assuming a bound R ≥ N − η for some η > independent of N , we have Tr h(cid:0) p + V (cid:1) γ (1) N i ≤ C (1 + β ) , (2.16) where C is a constant independent of β , N and R .Proof. We proceed in two steps.
Step 1.
Using a trial state u ⊗ N with u = | u | ∈ C ∞ c ( R ), we easily obtain from (1.11) andthe above bounds (note that the R -divergent mixed two-body term is zero on such a u , andthat the singular two-body term gives a lower-order contribution) E R ( N ) ≤ C (1 + β ) N. (2.17)Next we use the diamagnetic inequality [29, Theorem 7.21] in each variable to obtain h Ψ N , H RN Ψ N i = N X j =1 Z R N (cid:16)(cid:12)(cid:12)(cid:0) − i ∇ j + α A Rj (cid:1) Ψ N (cid:12)(cid:12) + V ( x j ) | Ψ N | (cid:17) dx . . . dx N ≥ N X j =1 Z R N (cid:16) |∇ j | Ψ N || + V ( x j ) | Ψ N | (cid:17) dx . . . dx N . We deduce the bound Tr h(cid:0) p + V (cid:1) γ (1) N, + i ≤ C (1 + β ) , (2.18)where we denote γ ( k ) N, + := Tr k +1 → N [ | | Ψ N | i h | Ψ N | | ]the reduced k -body density matrix of | Ψ N | . Step 2.
Next we expand the Hamiltonian and use the Cauchy-Schwarz inequality foroperators to obtain H RN = N X j =1 (cid:0) p j + αp j · A Rj + α A Rj · p j + α | A Rj | + V ( x j ) (cid:1) ≥ N X j =1 (cid:0) (1 − δ − ) p j + (1 − δ ) α | A Rj | + V ( x j ) (cid:1) = N X j =1 (cid:18)
12 ( p j + V ( x j )) − β ( N − | A Rj | (cid:19) , choosing δ = 4. Thus, using (2.17) we haveTr h(cid:0) p + V (cid:1) γ (1) N i ≤ C (1 + β ) + Cβ N ( N − * Ψ N , N X j =1 | A Rj | Ψ N + . (2.19)Then, since the last term in the right-hand side is purely a potential term * Ψ N , N X j =1 | A Rj | Ψ N + = * | Ψ N | , N X j =1 | A Rj | | Ψ N | + . We then expand the squares as in (1.11), and use Lemmas 2.2 and 2.4 to obtain for any ε > N ( N − * | Ψ N | , N X j =1 | A Rj | | Ψ N | + ≤ C Tr h ∇ ⊥ w R ( x − x ) · ∇ ⊥ w R ( x − x ) γ (3) N, + i + CN − Tr h |∇ w R ( x − x ) | γ (2) N, + i ≤ C Tr h ( p + 1) ⊗ ⊗ γ (3) N, + i + C ε R − ε N − Tr h ( p + 1) ⊗ γ (2) N, + i ≤ C (cid:0) C ε N − R − ε (cid:1) Tr h ( p + 1) γ (1) N, + i . Inserting the estimate (2.18) and recalling that we assume R ≥ N − η we conclude the proofby going back to (2.19). (cid:3) Mean-field limit
We now turn to the study of the mean-field limit per se. The strategy is the same asin [26], but the peculiarities of the anyon Hamiltonian add some important twists, and weshall rely heavily on the estimates of the preceding section.3.1.
Preliminaries.
We first recall some constructions from [24, 26].
Energy cut-off.
We denote by P the spectral projector of − ∆ + V below a given (large)energy cut-off Λ that we shall optimize over in the end: P := h ≤ Λ , h = − ∆ + V. (3.1)Let N Λ = dim( P L ( R ))be the number of energy levels obtained this way, and recall the following Cwikel-Lieb-Rozenblum type inequality, proved by well-known methods, as in [26, Lemma 3.3]: Lemma 3.1 ( Number of energy levels below the cut-off ) . For Λ large enough we have N Λ ≤ C Λ /s . (3.2)We shall also denote Q = − P the orthogonal projector onto excited energy levels. VERAGE FIELD APPROXIMATION 17
Localization in Fock space.
We quickly recall the procedure of geometric localization,following the notation of [24]. Let γ N be an arbitrary N -body (mixed) state. Associatedwith the given projector P , there is a localized state G PN in the Fock space F ( H ) = C ⊕ H ⊕ H ⊕ · · · of the form G PN = G PN, ⊕ G PN, ⊕ · · · ⊕ G PN,N ⊕ ⊕ · · · (3.3)with the property that its reduced density matrices satisfy P ⊗ n γ ( n ) N P ⊗ n = (cid:0) G PN (cid:1) ( n ) = (cid:18) Nn (cid:19) − N X k = n (cid:18) kn (cid:19) Tr n +1 → k (cid:2) G PN,k (cid:3) (3.4)for any 0 ≤ n ≤ N . Here we use the convention that γ ( n ) N := Tr n +1 → N [ γ N ] , which differs from the convention of [24], whence the different numerical factors in (3.4). Wealso have a localized state G QN corresponding to the projector Q , which is defined similarly.The relations (3.4) determine the localized states G PN , G QN uniquely and they ensurethat G PN and G QN are (mixed) states on the projected Fock spaces F ( P H ) and F ( Q H ),respectively: N X k =0 Tr h G P/QN,k i = 1 . (3.5) de Finetti measure for the projected state. We will apply the quantitative de FinettiTheorem in finite dimensional spaces of [8, 5, 17, 27] to the localized state G PN , in or-der to approximate its three-body density matrix. The following is the equivalent of [26,Lemma 3.4] and the proof is exactly similar: Lemma 3.2 ( Quantitative quantum de Finetti for the localized state ) . Let γ N be an arbitrary N -body (mixed) state. Define dµ N ( u ) := N X k =3 (cid:18) N (cid:19) − (cid:18) k (cid:19) dµ N,k ( u ) , dµ N,k ( u ) := dim( P H ) k sym D u ⊗ k , G PN,k u ⊗ k E du (3.6) and e γ (3) N := Z SP H | u ⊗ ih u ⊗ | dµ N ( u ) . (3.7) Then there is a constant
C > such that for every N ∈ N and Λ > , we have Tr (cid:12)(cid:12)(cid:12) P ⊗ γ (3) N P ⊗ − e γ (3) N (cid:12)(cid:12)(cid:12) ≤ CN Λ N . (3.8)
Truncated Hamiltonian.
For an energy lower bound we are first going to roughlybound some terms in the Hamiltonian. Let us introduce the effective three-body Hamilton-ian˜ H R := 13 ( h + h + h ) + β X ≤ j = k ≤ (cid:16) p j · ∇ ⊥ w R ( x j − x k ) + ∇ ⊥ w R ( x j − x k ) · p j (cid:17) + β ∇ ⊥ w R ( x − x ) · ∇ ⊥ w R ( x − x ) (3.9)where h i is understood to act on the i -th variable (recall that h = − ∆ + V ). For shortnesswe denote W = p · ∇ ⊥ w R ( x − x ) + ∇ ⊥ w R ( x − x ) · p the two-body part of ˜ H R , and W = ∇ ⊥ w R ( x − x ) · ∇ ⊥ w R ( x − x )its three-body part. With this notation˜ H R := 13 ( h + h + h ) + β X ≤ i = j ≤ W ( i, j ) + β W where W ( i, j ) acts on variables i and j . Also note that for k u k = 1, by (1.14), (1.16), h u ⊗ , ˜ H R u ⊗ i = E af R [ u ] ≥ E af R . We bound the full energy from below in terms of a projected version of ˜ H R : Proposition 3.3 ( Truncated three-body Hamiltonian ) . Let Ψ N be a (sequence of ) approximate ground state(s) for H RN with associated reduceddensity matrices γ ( k ) N . Then, for any ε > and R small enough, N h Ψ N , H RN Ψ N i ≥ Tr h ˜ H R P ⊗ γ (3) N P ⊗ i + C β Λ Tr[ Qγ (1) N ] − C β (cid:18) N + C ε √ Λ R ε + 1Λ R (cid:19) . (3.10) Proof.
We proceed in several steps.
Step 1.
We first claim that1 N h Ψ N , H RN Ψ N i ≥ Tr h ˜ H R γ (3) N i − C β N − . (3.11)To see this, we start from (1.11). For a lower bound we drop the term on the fourth line,which is positive. Then one only has to correct the N -dependent factors in front of thethird line. The term we have to drop to obtain (3.11) is bounded as β (cid:12)(cid:12)(cid:12)(cid:12) − N − N − (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) Tr h(cid:16) ∇ ⊥ w R ( x − x ) · ∇ ⊥ w R ( x − x ) (cid:17) γ (3) N i(cid:12)(cid:12)(cid:12) ≤ C β N − upon using the a priori bound (2.16) combined with (2.10). Step 2.
We next proceed to bound the right-hand side of (3.11) from below in terms ofa localized version of ˜ H R and remainder terms to be estimated in the next step. We shall VERAGE FIELD APPROXIMATION 19 need the projectors Π = ⊗ − P ⊗ Π = ⊗ − P ⊗ and make a repeated use of the inequality ABC + CBA ≥ − εA | B | A − ε − C | B | C, ε > , (3.12)for any self-adjoint operators A, B, C .We claim thatTr h ˜ H R γ (3) N i ≥ Tr h ˜ H R P ⊗ γ (3) N P ⊗ i + Tr h hQγ (1) N Q i − | β | (3 + δ ) Tr h Π | W | Π γ (2) N i − | β | δ − Tr h P ⊗ | W | P ⊗ γ (2) N i − | β | Tr h P ⊗ ⊗ Q | W (1 , | P ⊗ ⊗ Qγ (3) N i − β (1 + δ ) Tr h Π | W | Π γ (3) N i − β δ − Tr h P ⊗ | W | P ⊗ γ (3) N i (3.13)where δ and δ are two positive parameters to be chosen later on.To prove (3.13), first note thatTr h ˜ H R γ (3) N i = Tr h hγ (1) N i + β h W γ (2) N i + β Tr h W γ (3) N i . Then, for the one-body term we haveTr h hγ (1) N i = Tr h P hP γ (1) N i + Tr h QhQγ (1) N i ≥
13 Tr h P ⊗ ( h + h + h ) P ⊗ γ (3) N i + Tr h QhQγ (1) N i using that h commutes with P and Q , P Q = QP = 0 and the fact that h is a positiveoperator.For the two-body term we writeTr h W γ (2) N i = Tr h P ⊗ W (1 , P ⊗ γ (3) N i + Tr h Π W (1 , γ (3) N i + Tr h(cid:0) P ⊗ W (1 , + Π W (1 , P ⊗ (cid:1) γ (3) N i . Next, since Π = P ⊗ ⊗ Q + Π ⊗ P + Π ⊗ Q, (3.14)and W (1 ,
2) only acts on the first two variables, this simplifies intoTr h W γ (2) N i = Tr h P ⊗ W (1 , P ⊗ γ (3) N i + Tr h Π W (1 , γ (3) N i + Tr h(cid:0) P ⊗ W (1 , ⊗ P + Π ⊗ P W (1 , P ⊗ (cid:1) γ (3) N i ≥ Tr h P ⊗ W (1 , P ⊗ γ (3) N i + Tr h Π W (1 , γ (3) N i − δ Tr h Π ⊗ P | W (1 , | Π ⊗ P γ (3) N i − δ − Tr h P ⊗ | W (1 , | P ⊗ γ (3) N i where we use (3.12) to obtain the lower bound. Then, using (3.14) and (3.12) again for thesecond term of the right-hand side, as well as P, Q ≤ , we getTr h W γ (2) N i ≥ Tr h P ⊗ W (1 , P ⊗ γ (3) N i − δ − Tr h P ⊗ | W | P ⊗ γ (2) N i − (3 + δ ) Tr h Π | W | Π γ (2) N i − h P ⊗ ⊗ Q | W (1 , | P ⊗ ⊗ Qγ (3) N i . Finally, the three-body term is dealt with similarly:Tr h W γ (3) N i = Tr h(cid:0) P ⊗ + Π (cid:1) W (cid:0) P ⊗ + Π (cid:1) γ (3) N i ≥ Tr h P ⊗ W P ⊗ γ (3) N i − (1 + δ ) Tr h Π | W | Π γ (3) N i − δ − Tr h P ⊗ | W | P ⊗ γ (3) N i using (3.12) again. All in all, using also the symmetry of γ (3) N , we obtain (3.13). Step 3.
We next estimate the remainder terms in (3.13). First we note thatTr h hQγ (1) N Q i ≥ Λ2 Tr h Qγ (1) N Q i + √ Λ2 Tr h √ hQγ (1) N Q i ≥ Λ4 Tr h Qγ (1) N Q i + Λ20 Tr h Π γ (3) N Π i + √ Λ4 Tr h √ h Π γ (2) N Π i . (3.15)The first inequality is just the definition of Q , and to see the second one we first write2 Tr h √ hQγ (1) N Q i = Tr h √ h Q ⊗ γ (2) N i + Tr h √ h ⊗ Qγ (2) N i = Tr h √ h ( Q ⊗ P + Q ⊗ Q ) γ (2) N i + Tr h √ h ( P ⊗ Q + Q ⊗ Q ) γ (2) N i = Tr h √ h ( Q ⊗ P + Q ⊗ Q ) γ (2) N ( P ⊗ + Π ) i + Tr h √ h ( P ⊗ Q + Q ⊗ Q ) γ (2) N ( P ⊗ + Π ) i = Tr h √ h Π γ (2) N Π i + Tr h √ h Q ⊗ Qγ (2) N Q ⊗ Q i + Tr h(cid:16) √ h − √ h (cid:17) P ⊗ Qγ (2) N P ⊗ Q i ≥ Tr h √ h Π γ (2) N Π i , where we use repeatedly the cyclicity of the trace and the fact that √ h commutes with P and Q , along with the fact that P Q = QP = 0 andΠ = ⊗ − P ⊗ = Q ⊗ Q + P ⊗ Q + Q ⊗ P. In the last step we also use that as operators √ h P ⊗ Q ≥ √ Λ P ⊗ Q ≥ √ h P ⊗ Q VERAGE FIELD APPROXIMATION 21 by definition of the projectors P and Q . This gives the third term in the right-hand sideof (3.15). The second one arises from similar considerations:Tr h Π γ (3) N i = Tr h Π ⊗ Qγ (3) N i + Tr h P ⊗ ⊗ Qγ (3) N i + Tr h Π ⊗ P γ (3) N i ≤ h Qγ (1) N i + Tr h Π γ (2) N i = 2 Tr h Qγ (1) N i + Tr h ( P ⊗ Q + Q ⊗ P + Q ⊗ Q ) γ (2) N i ≤ h Qγ (1) N i . Next, using (2.6), we haveTr h Π | W | Π γ (2) N i ≤ CR Tr h | p | Π γ (2) N Π i ≤ CR Tr h √ h Π γ (2) N Π i (3.16)by operator monotonicity of the square-root, and by (2.1)Tr h Π | W | Π γ (3) N i ≤ CR Tr h Π γ (3) N Π i . (3.17)Moreover, using (2.6) again we getTr h P ⊗ ⊗ Q | W (1 , | P ⊗ ⊗ Qγ (3) N i ≤ C √ Λ R Tr h P ⊗ ⊗ Qγ (3) N i ≤ C √ Λ R Tr h Qγ (1) N i so that, combining with (3.15), choosing for some small fixed c , c > δ = c √ Λ R, δ = c Λ R and Λ large enough (i.e. Λ R > c for c large enough), we getTr h hQγ (1) N Q i − | β | (3 + δ ) Tr h Π | W | Π γ (2) N i − | β | Tr h P ⊗ ⊗ Q | W (1 , | P ⊗ ⊗ Qγ (3) N i − β (1 + δ ) Tr h Π | W | Π γ (3) N i ≥ C Tr h hQγ (1) N Q i for some fixed constant C >
0. Then, inserting in (3.13), we deduceTr h ˜ H R γ (3) N i ≥ Tr h ˜ H R P ⊗ γ (3) N P ⊗ i + C Tr h hQγ (1) N Q i − C √ Λ R Tr h P ⊗ | W | P ⊗ γ (2) N i − C Λ R Tr h P ⊗ | W | P ⊗ γ (3) N i . (3.18)But, using (2.7), (2.16), and Tr γ ( k ) N = 1,Tr h P ⊗ | W | P ⊗ γ (2) N i ≤ C ε R − ε Tr h P ⊗ ( p + 1) P ⊗ γ (2) N i ≤ C ε R − ε Tr h ( p + 1) γ (1) N i ≤ C ε R − ε , whereas, using (2.10) and (2.16) againTr h P ⊗ | W | P ⊗ γ (3) N i ≤ C Tr h P ⊗ ( p + 1) P ⊗ γ (3) N i ≤ C Tr h ( p + 1) γ (1) N i ≤ C, which completes the proof. (cid:3) Energy bounds.
In this subsection we prove the energy bounds establishing (1.19).The upper bound is obtained as usual by testing against a factorized trial state. Namely,taking Ψ N = ( u af R ) ⊗ N in (1.11) with u af R a normalized minimizer of E af R , and using (1.14),(1.16), Lemmas 2.2, 2.3, 2.4, and the diamagnetic inequality (A.3), one finds E R ( N ) N ≤ E af R [ u af R ] + ( E af R [ u af R ] + 1) (cid:18) Cβ N + C ε β R − ε N (cid:19) = E af R + o (1) → E af , (3.19)as R ∼ N − η with N → ∞ , where we also used Proposition A.6. Note that for this upperbound we can allow any rate 0 < η < ∞ , and may even take β = β ( N ) → ∞ .For the lower bound, inserting (3.8) in the estimate of Proposition 3.3 we get for anysequence of ground states Ψ N that1 N h Ψ N , H RN Ψ N i ≥ Tr h ˜ H R e γ (3) N i + C Λ Tr[ Qγ (1) N ] − C N Λ N (cid:13)(cid:13)(cid:13) P ⊗ ˜ H R P ⊗ (cid:13)(cid:13)(cid:13) − C (cid:18) N + C ε √ Λ R ε + 1Λ R (cid:19) ≥ Tr h ˜ H R e γ (3) N i + C Λ Tr[ Qγ (1) N ] − C Λ /s N (1 + | log R | ) − C (cid:18) N + C ε √ Λ R ε + 1Λ R (cid:19) . (3.20)We have here used Estimates (2.8) and (2.10) from Subsection 2.1, along with P p P ≤ P hP ≤ Λto bound the operator norm of P ⊗ ˜ H R P ⊗ and Lemma 3.1 to bound N Λ . Main term.
Since by definition e γ (3) N is a superposition of tensorized states we getTr h ˜ H R e γ (3) N i ≥ E af R Tr he γ (3) N i . We then denote λ := Tr h P γ (1) N i = N X k =0 kN Tr (cid:2) G PN,k (cid:3) (3.21)the fraction of P -localized particles. Using the simple estimate (cid:12)(cid:12)(cid:12)(cid:12) kN k − N − k − N − − k N (cid:12)(cid:12)(cid:12)(cid:12) ≤ CN − for 0 ≤ k ≤ N, VERAGE FIELD APPROXIMATION 23 it follows from (3.5), (3.6), (3.7), and Jensen’s inequality thatTr he γ (3) N i = Z SP H dµ N = N X k =3 (cid:18) N (cid:19) − (cid:18) k (cid:19) Tr (cid:2) G PN,k (cid:3) ≥ N X k =0 (cid:18) kN (cid:19) Tr (cid:2) G PN,k (cid:3) − O ( N − ) ≥ λ − O ( N − ) . (3.22)Since on the other hand Tr h Qγ (1) N i = 1 − λ, we concludeTr h ˜ H R e γ (3) N i + C Λ Tr[ Qγ (1) N ] ≥ λ E af R + C Λ(1 − λ ) − CN − ≥ E af R − CN − . (3.23)For the last inequality we bound from below in terms of the infimum with respect to 0 ≤ λ ≤
1. Since Λ is a very large number and E af R is bounded as R → λ = 1. Optimizing the error.
We next choose Λ to minimize the error in (3.20). We assumethat R behaves at worst as R ∼ N − η . Changing a little bit η if necessary we may ignore the | log R | and R ε factors, and we thushave to minimize 1 √ Λ R + 1Λ R + Λ /s N .
We pick Λ = N (1+ η )(1+4 / s ) − to equate the first and the last term and get1 √ Λ R + 1Λ R + Λ /s ) N = O ( N (1+ η )(1+1 /s )(1+4 / s ) − − ) + O ( N (1+ η )(1+1 /s )(1+4 / s ) − − ) , and this is small provided η < η := 14 (cid:18) s (cid:19) − . Since this is the main error term we conclude that the lower bound corresponding to (1.19)holds provided R ∼ N − η with η < η , as stated in the theorem. The limit E af R → E af isdealt with in Appendix A.3.4. Convergence of states.
Given the previous constructions and energy estimates, theproof of (1.20) follows almost exactly [26, Section 4.3] and is thus only sketched.Modulo extraction of subsequences we have γ ( k ) N ⇀ ∗ γ ( k )4 D. LUNDHOLM AND N. ROUGERIE weakly- ∗ in the trace-class as N → ∞ . From Proposition 2.6 we know that ( − ∆ + V ) γ (1) N is uniformly bounded in trace-class. Under our assumptions, ( − ∆ + V ) − is compact andwe may thus, modulo a further extraction, assume that γ (1) N −→ N →∞ γ (1) strongly in trace-class norm. Then, by [25, Corollary 2.4], we also have γ ( k ) N −→ N →∞ γ ( k ) strongly for all k ≥ µ N defined in Lemma 3.2 converges (modulo extraction)to a limit probability measure µ ∈ P ( S H ) on the unit sphere of H = L ( R ) and that γ ( k ) = Z u ∈ S H | u ⊗ k ih u ⊗ k | dµ ( u ) for all k ≥ . (3.24)To see this, we first apply (3.8), with the above choice of Λ. We obtainTr (cid:12)(cid:12)(cid:12)(cid:12) P ⊗ γ (3) N P ⊗ − Z SP H | u ⊗ ih u ⊗ | dµ N ( u ) (cid:12)(cid:12)(cid:12)(cid:12) → . (3.25)On the other hand, combining (3.23) with the energy upper bound (3.19) we get λ → λ is the fraction of P -localized particles defined in (3.21). Using Jensen’s inequalityas in (3.22) we deduce that µ N ( SP H ) = Tr h P ⊗ γ (3) N P ⊗ i → . (3.26)Combining with (3.25) yieldsTr (cid:12)(cid:12)(cid:12)(cid:12) γ (3) N − Z SP H | u ⊗ ih u ⊗ | dµ N ( u ) (cid:12)(cid:12)(cid:12)(cid:12) → . (3.27)Testing this with a sequence of finite rank orthogonal projectors P K −→ K →∞ and using the strong convergence of γ (3) N giveslim K →∞ lim N →∞ µ N ( P K H ) = 1 , and we obtain the existence of a limit measure µ supported on the unit ball of H by atightness argument. Then (3.24) for k = 3 follows from (3.27). Since γ (3) N converges strongly,the limit has trace 1 and µ must be supported on the unit sphere. Obtaining (3.24) forlarger k is a general argument based on (3.26). We refer to [26, Section 4.3] for details.There only remains to prove that µ is supported on M af . But it follows easily fromcombining previously obtained energy bounds that Z SP H (cid:12)(cid:12)(cid:12) E af R − E af R [ u ] (cid:12)(cid:12)(cid:12) dµ N ( u ) −→ N →∞ . VERAGE FIELD APPROXIMATION 25
Using in addition the results of Appendix A, in particular Proposition A.6, we obtain for alarge but fixed constant
C > Z E af [ u ] ≤ C (cid:12)(cid:12)(cid:12) E af − E af [ u ] (cid:12)(cid:12)(cid:12) dµ N ( u ) −→ N →∞ Z E af [ u ] ≥ C dµ N ( u ) −→ N →∞ . Then clearly µ must be supported on M af , which concludes the proof. (cid:3) Appendix A. Properties of the average-field functional
In this appendix we etablish some of the fundamental properties of the functional (1.10)and its limit R → β ∈ R and V : R → R + we define the average-field energy functional E af [ u ] := Z R (cid:16)(cid:12)(cid:12)(cid:0) ∇ + iβ A [ | u | ] (cid:1) u (cid:12)(cid:12) + V | u | (cid:17) , (A.1)with the self-generated magnetic potential A [ ρ ] := ∇ ⊥ w ∗ ρ = Z R ( x − y ) ⊥ | x − y | ρ ( y ) dy, curl A [ ρ ] = 2 πρ. The functional is certainly well-defined for u ∈ C ∞ c ( R ), but we should ask what its naturaldomain is. We then have to make a meaning of E af [ u ] for general u ∈ L ( R ) and theproblem is that it is not certain that A [ | u | ] ∈ L even though u ∈ L (see [12] for anexample ), so the product A [ | u | ] u may not be well-defined as a distribution (while ∇ u certainly is). One way around this is to reconsider the form of the functional when actingon regular enough functions such that we can write u = | u | e iϕ where ϕ is real. Then (cid:12)(cid:12) ( ∇ + iβ A [ | u | ]) u (cid:12)(cid:12) = (cid:12)(cid:12) ∇| u | + i | u | ( ∇ ϕ + β A [ | u | ]) (cid:12)(cid:12) = |∇| u || + (cid:12)(cid:12) | u |∇ ϕ + β A [ | u | ] | u | (cid:12)(cid:12) , where also ∇ ϕ = | u | − ℑ ¯ u ∇ u and ∇| u | = | u | − ℜ ¯ u ∇ u . Hence, an alternative definition isgiven by E af [ u ] := Z R |∇| u || + (cid:12)(cid:12)(cid:12)(cid:12) ℑ ¯ u | u | ∇ u + β A [ | u | ] | u | (cid:12)(cid:12)(cid:12)(cid:12) + V | u | ! , (A.2)and the advantage of this formulation is that it makes clear that we actually demand | u | ∈ H ( R ) in order for E af [ u ] < ∞ . We can then use the following lemma to see that infact A [ | u | ] u ∈ L ( R ), and hence also ∇ u ∈ L ( R ). (And conversely this also shows thatif A [ | u | ] u / ∈ L ( R ) then we have no chance of making sense out of E af [ u ].) Lemma A.1 ( Bound on the magnetic term ) . We have for any u ∈ L ( R ) that Z R (cid:12)(cid:12) A [ | u | ] (cid:12)(cid:12) | u | ≤ k u k L ( R ) Z R |∇| u || . Note that by Young’s inequality we have for any u ∈ L ( R ) that A [ | u | ] ∈ L p ( R ) + εL ∞ ( R ) for p ∈ [1 , Proof.
This follows from symmetry and from the three-body Hardy inequality of Lemma 2.5: Z R (cid:12)(cid:12) A [ | u | ]( x ) (cid:12)(cid:12) | u ( x ) | dx = Z Z Z R x − y | x − y | · x − z | x − z | | u ( x ) | | u ( y ) | | u ( z ) | dxdydz = 16 Z R R ( X ) (cid:12)(cid:12) | u | ⊗ (cid:12)(cid:12) dX ≤ Z R (cid:12)(cid:12) ∇ X | u | ⊗ (cid:12)(cid:12) dX = 32 Z R |∇| u || dx (cid:18)Z R | u | dx (cid:19) . (cid:3) We can therefore define the domain of E af to be (and otherwise let E af [ u ] := + ∞ ) D af := (cid:26) u ∈ H ( R ) : Z R V | u | < ∞ (cid:27) , and we find using Cauchy-Schwarz, Lemma A.1, and |∇| u || ≤ |∇ u | that for u ∈ D af ≤ E af [ u ] ≤ k∇ u k + 2 β k A [ | u | ] u k + Z V | u | ≤ (2 + 3 β k u k ) k∇ u k + Z V | u | < ∞ . The ground-state energy of the average-field functional is then given by E af := inf (cid:26) E af [ u ] : u ∈ D af , Z R | u | = 1 (cid:27) . For convenience we also make the assumption on V that V ( x ) → + ∞ as | x | → ∞ and that C ∞ c ( R ) ⊆ D af is a form core for k u k L V := R R V | u | , with − ∆ + V essentially self-adjointand with purely discrete spectrum (see, e.g., [46, Theorem XIII.67]). This is then also acore for E af : Proposition A.2 ( Density of regular functions in the form domain ) . C ∞ c ( R ) is dense in D af w.r.t. E af , namely for any u ∈ D af there exists a sequence ( u n ) n →∞ ⊂ C ∞ c ( R ) such that k u − u n k H → and E af [ u n ] → E af [ u ] as n → ∞ . Proof.
Take u ∈ D af , then k∇ u k L < ∞ and hence also k u k L p < ∞ for any p ∈ [2 , ∞ ) bySobolev embedding. We use that C ∞ c ( R ) is dense in H ( R ), so there exists a sequence( u n ) n →∞ ⊂ C ∞ c s.t. k u − u n k H →
0. Also, (cid:12)(cid:12)(cid:13)(cid:13) ( ∇ + iβ A [ | u | ]) u (cid:13)(cid:13) − (cid:13)(cid:13) ( ∇ + iβ A [ | u n | ]) u n (cid:13)(cid:13) (cid:12)(cid:12) ≤ (cid:13)(cid:13) ( ∇ + iβ A [ | u | ]) u − ( ∇ + iβ A [ | u n | ]) u n (cid:13)(cid:13) ≤ k∇ ( u − u n ) k + | β |k ( A [ | u | ] − A [ | u n | ]) u + A [ | u n | ]( u − u n ) k ≤ k u − u n k H + | β | (cid:13)(cid:13) A [ | u | − | u n | ] u (cid:13)(cid:13) + | β | (cid:13)(cid:13) A [ | u n | ]( u − u n ) (cid:13)(cid:13) , where by H¨older’s and generalized Young’s inequalities (cid:13)(cid:13) A [ | u | − | u n | ] u (cid:13)(cid:13) ≤ (cid:13)(cid:13) A [ | u | − | u n | ] (cid:13)(cid:13) k u k ≤ C (cid:13)(cid:13) | u | − | u n | (cid:13)(cid:13) / k∇ w k ,w k u k ≤ C ′ k u − u n k / ≤ C ′′ k u − u n k H → , and similarly (cid:13)(cid:13) A [ | u n | ]( u − u n ) (cid:13)(cid:13) ≤ C k u − u n k H → , as n → ∞ .We also have continuity for k u k L V here since we assumed that C ∞ c ( R ) is a form core. (cid:3) VERAGE FIELD APPROXIMATION 27
Lemma A.3 ( Basic magnetic inequalities ) . We have for u ∈ D af that (diamagnetic inequality) Z R (cid:12)(cid:12) ( ∇ + iβ A [ | u | ]) u (cid:12)(cid:12) ≥ Z R |∇| u || , (A.3) and Z R (cid:12)(cid:12) ( ∇ + iβ A [ | u | ]) u (cid:12)(cid:12) ≥ π | β | Z R | u | . (A.4) Proof.
By density we can w.l.o.g. assume u ∈ C ∞ c ( R ). We then have A [ | u | ] ∈ C ∞ ( R ) ⊆ L ( R ) and hence the first inequality follows by the usual diamagnetic inequality (see e.g.Theorem 2.1.1 in [14]). Furthermore, by e.g. Lemma 1.4.1 in [14], Z R (cid:12)(cid:12) ( ∇ + iβ A [ | u | ]) u (cid:12)(cid:12) ≥ ± Z R curl (cid:0) β A [ | u | ] (cid:1) | u | , which proves the second inequality since curl A [ | u | ] = 2 π | u | . Instead of using densitywe could also have used the formulation (A.2) or the fact that u ∈ H ⇒ A [ | u | ] ∈ L p , p ∈ (2 , ∞ ) by generalized Young. (cid:3) Proposition A.4 ( Existence of minimizers ) . For any value of β ∈ R there exists u af ∈ D af with R R | u af | = 1 and E af [ u af ] = E af .Proof. First note that for u ∈ D af , by Lemma A.1 and Lemma A.3, k∇ u k = (cid:13)(cid:13) ∇ u + iβ A [ | u | ] u − iβ A [ | u | ] u (cid:13)(cid:13) ≤ E af [ u ] / + | β | (cid:13)(cid:13) A [ | u | ] u (cid:13)(cid:13) ≤ E af [ u ] / + | β | r k u k k∇| u |k ≤ | β | r k u k ! E af [ u ] / . Now take a minimizing sequence( u n ) n →∞ ⊂ D af , k u n k = 1 , lim n →∞ E af [ u n ] = E af . Then clearly ( u n ) is uniformly bounded in both L ( R ), L V , and H ( R ) (and hence in L p ( R ), p ∈ [2 , ∞ )), and therefore by the Banach-Alaoglu theorem there exists u af ∈ D af and a weakly convergent subsequence (still denoted u n ) such that u n ⇀ u af in L ( R ) ∩ L V ∩ L p ( R ) , ∇ u n ⇀ ∇ u af in L ( R ) . Moreover, since ( − ∆ + V + 1) − / is compact we have that u n = ( − ∆ + V + 1) − / ( − ∆ + V + 1) / u n is actually strongly convergent (again extracting a subsequence), hence u n → u af in L ( R ) . Also, A [ | u n | ] converges pointwise a.e. to A [ | u | ] by weak convergence of u n in L p and, bythe trick of Lemma A.1, (cid:13)(cid:13) A [ | u n | ] u n (cid:13)(cid:13) = 16 Z R R ( X ) − (cid:12)(cid:12) | u n | ⊗ (cid:12)(cid:12) dX → Z R R ( X ) − (cid:12)(cid:12) | u | ⊗ (cid:12)(cid:12) dX = (cid:13)(cid:13) A [ | u | ] u (cid:13)(cid:13)
228 D. LUNDHOLM AND N. ROUGERIE by dominated convergence. The functions A [ | u n | ] u n are therefore even strongly convergingto A [ | u | ] u in L ( R ) by dominated convergence. It then follows that (cid:13)(cid:13) ( ∇ + iβ A [ | u | ]) u (cid:13)(cid:13) = sup k v k =1 |h∇ u + iβ A [ | u | ] u, v i| = sup k v k =1 lim n →∞ |h∇ u n + iβ A [ | u n | ] u n , v i|≤ lim inf n →∞ sup k v k =1 |h∇ u n + iβ A [ | u n | ] u n , v i| = lim inf n →∞ (cid:13)(cid:13) ( ∇ + iβ A [ | u n | ]) u n (cid:13)(cid:13) , and since k·k L V is also weakly lower semicontinuous (see, e.g., [44, Supplement to IV.5]),we have lim inf n →∞ E af [ u n ] ≥ E af [ u af ]. Thus, with k u af k = lim n →∞ k u n k = 1, we also have E af [ u af ] = E af . (cid:3) Proposition A.5 ( Convergence to bosons ) . Let E resp. u denote the ground-state eigenvalue resp. normalized eigenfunction of thenon-magnetic Schr¨odinger operator H = − ∆ + V , with V ∈ L ∞ loc . We have E af( β ) → β → E , and that given an arbitrary sequence ( u β ) of minimizers for E af( β ) u β → β → u in L ( R ) up to a subsequence and a constant phase.Proof. Note that under our conditions for V , u ∈ D af is the unique minimizer of E = E af( β =0) and can be taken positive (see, e.g., [29, Theorem 11.8]). By the diamagnetic inequality(A.3), and by taking the trial state u = | u | in E af( β =0) , we find E ≤ E af( β ) ≤ E af( β ) [ u ] = E [ u ] + β (cid:13)(cid:13) A [ | u | ] u (cid:13)(cid:13) ≤ (1 + Cβ ) E (where we also used Lemma A.1), and hence E af( β ) → E as β →
0. Now consider a sequence( u β ) ⊂ D af of minimizers as β → E af [ u β ] → E , k u β k = 1. Then, because of uniformboundedness and as in the proof of Proposition A.4, we have after taking a subsequencethat u β → u for some u ∈ D af , k u k = 1, and also k∇ u k = sup k v k =1 |h∇ u, v i| = sup k v k =1 lim β → (cid:12)(cid:12) h∇ u β + iβ A [ | u β | ] u β , v i (cid:12)(cid:12) ≤ lim inf β → (cid:13)(cid:13) ∇ u β + iβ A [ | u β | ] u β (cid:13)(cid:13) , so E ≤ E [ u ] ≤ lim inf β → E af( β ) [ u β ] . It follows that E [ u ] = E and hence u = u up to a constant phase. (cid:3) VERAGE FIELD APPROXIMATION 29
From the bound (A.4) we observe that the self-generated magnetic interaction is strongerthan a contact interaction of strength 2 π | β | (despite the fact that we already removed asingular repulsive interaction in the initial regularization step for extended anyons). Hencewe have not only E af ≥ E by the diamagnetic inequality, but also E af ≥ min ρ ≥ , R R ρ =1 Z R (cid:0) π | β | ρ + V ρ (cid:1) , (A.5)which can be computed for given V by straightforward optimization.Let us now consider the corresponding situation for the regularized functional (extendedanyons) E af R [ u ] := Z R (cid:16)(cid:12)(cid:12)(cid:0) ∇ + iβ A R [ | u | ] (cid:1) u (cid:12)(cid:12) + V | u | (cid:17) , A R [ ρ ] := ∇ ⊥ w R ∗ ρ, R > . Since ∇ w R ∈ L ∞ ( R ) we have A R [ | u | ] ∈ L ∞ ( R ) with (cid:13)(cid:13) A R [ | u | ] (cid:13)(cid:13) ∞ ≤ CR k u k and instead of Lemma A.1 we have (cid:13)(cid:13) A R [ | u | ] u (cid:13)(cid:13) ≤ C k u k k| u |k H using Lemma 2.4. Hence the natural domain is again D af and all properties establishedabove for E af are also found to be valid for E af R (except (A.4) and (A.5) which now haveregularized versions). Denoting E af R := min {E af R [ u ] : u ∈ D af , k u k = 1 } , we furthermore have the following relationship: Proposition A.6 ( Convergence to point-like anyons ) . The functional E af R converges pointwise to E af as R → . More precisely, for any u ∈ D af (cid:12)(cid:12)(cid:12) E af R [ u ] − E af [ u ] (cid:12)(cid:12)(cid:12) ≤ C u | β | (1 + β )(1 + E af [ u ]) / R, (A.6) where C u depends only on k u k . Hence, E af R → R → E af , and if ( u R ) R → ⊂ D af denotes a sequence of minimizers of E af R , then there exists a subse-quence ( u R ′ ) R ′ → s.t. u R ′ → u af as R ′ → , where u af is some minimizer of E af .Proof. We have for any u ∈ D af that (cid:12)(cid:12)(cid:13)(cid:13) ( ∇ + iβ A [ | u | ]) u (cid:13)(cid:13) − (cid:13)(cid:13) ( ∇ + iβ A R [ | u | ]) u (cid:13)(cid:13) (cid:12)(cid:12) ≤ (cid:13)(cid:13) ( ∇ + iβ A [ | u | ]) u − ( ∇ + iβ A R [ | u | ]) u (cid:13)(cid:13) = | β | (cid:13)(cid:13) ( A [ | u | ] − A R [ | u | ]) u (cid:13)(cid:13) ≤ | β | (cid:13)(cid:13) A [ | u | ] − A R [ | u | ] (cid:13)(cid:13) k u k = | β | (cid:13)(cid:13) ( ∇ w − ∇ w R ) ∗ | u | (cid:13)(cid:13) k u k , where by Young (cid:13)(cid:13) ( ∇ w − ∇ w R ) ∗ | u | (cid:13)(cid:13) ≤ k∇ w − ∇ w R k (cid:13)(cid:13) | u | (cid:13)(cid:13) ≤ k∇ w k L ( B (0 ,R )) k u k → , as R →
0, since ∇ w ∈ L ( R ). We deduce (A.6) by combining this with previousestimates of this appendix and Sobolev embeddings. It follows that E af R [ u ] → E af [ u ] as R → u R ) R → denote a sequence of minimizers of E af R : E af R = E af R [ u R ] , k u R k = 1 , and take u ∈ D af an arbitrary minimizer of E af . Then, since E af R ≤ E af R [ u ] → R → E af [ u ] = E af , we have that E af R is uniformly bounded as R → R → E af R ≤ E af . Then E af R [ u R ], and hence also E af [ u R ] ≤ C (cid:16) k u R k H + k u R k L V (cid:17) ≤ C ′ ( E af R [ u R ] + 1) , are uniformly bounded as well. As in the proof of Proposition A.4, there then exists astrongly convergent subsequence ( u R ′ ) R ′ → , with u R ′ → u ∈ D af . Also, by weak lowersemicontinuity E af [ u ] ≤ lim inf R ′ → E af [ u R ′ ], so that for any ε > R ′ > E af ≤ E af [ u ] ≤ E af [ u R ′ ] + ε ≤ E af R ′ [ u R ′ ] + 2 ε = E af R ′ + 2 ε, where we also used that the convergence is uniform for our uniformly bounded sequence u R by the bound (A.6). It follows that E af ≤ E af [ u ] ≤ E af + 3 ε , and hence u is a minimizerwith k u k = 1 and E af = E af [ u ] = lim R → E af R . (cid:3) References [1]
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KTH Royal Institute of Technology, Department of Mathematics, SE-100 44 Stockholm,Sweden
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