Homogenization of point interactions

HOMOGENIZA TION OF POINT INTERA CTIONS DOMENICO CAFIER O, MICHELE CORREGGI, AND DA VIDE FERMI Abstract. W e consider a non-relativistic quantum particle in R d , d = 2 or d = 3, interacting with singular zero-range p oten tials concentrated on a large collection of p oints. W e analyze the homogenization regime where the in tensities of the singular potentials and the distances b etw een the p oints simultaneously go to zero as their num ber grows, while the total in teraction strength remains finite. Assuming that the singular p oten tials ha v e negativ e scattering lengths and are uniformly distributed, we prov e the strong resolven t con vergence as N → ∞ of the family of op erators to a Sc hr¨ odinger op erator with a regular electrostatic p oten tial. The result is obtained via Γ-con verge of the asso ciated quadratic forms. Moreo ver, in presence of an external trapping p otential, the conv ergence is lifted to uniform resolv ent sense. Contents 1. In tro duction 1 2. Setting and Main Results 3 3. Preliminaries 7 3.1. Γ − con v ergence 8 3.2. Measures and distribution of centers 9 3.3. Green functions 10 3.4. Hilb ert scales asso ciated to H 0 11 4. Pro ofs 11 4.1. Pro of of the Γ − lim inf inequalit y 12 4.2. Pro of of the Γ − lim sup (in)equality 17 4.3. Pro of of the uniform resolven t conv ergence 19 References 20 1. Introduction Zero-range p otentials were in tro duced in the mid ‘30s to mo del the scattering of low-energy neutrons b y atomic nuclei and to describ e related atomic systems [BP35, Th35, F e36]. In this con text, “p otentials” of Dirac-delta type provide an effectiv e approximation whenever the particle wa v elength greatly exceeds the range of v ery strong forces. The first rigorous construction of a p oin t in teraction as a singular perturbation of the free Laplacian was given by Berezin and F addeev in 1961 [BF61]. Since then the mathematical description of zero-range p otentials has b een extensiv ely studied within the framework of self-adjoint extensions of symmetric op erators, using classical to ols such as von Neumann theory , quadratic forms, Krein-t yp e resolven t formulas, and b oundary triplets. Notably , such singular interactions can b e realized as renormalized limits of resonan t short-range p otentials. F or comprehensiv e exp ositions, we refer to the monographs [AGH+88, AK00] and the references therein. In this w ork, we study the quantum dynamics of a non-relativistic particle moving in a smo oth electro- magnetic background and interacting with several p oin t-lik e scatterers. F ormally , the system is describ ed 2020 Mathematics Subje ct Classific ation. 35J10, 35P05, 47A07, 47B93, 81Q10, 35B27. Key words and phr ases. p oint in teractions, zero-range p otentials, quadratic forms, Γ-conv ergence, resolv ent conv ergence. 1 2 DOMENICO CAFIERO, MICHELE CORREGGI, AND D A VIDE FERMI b y a Schr¨ odinger op erator of the form ( − i ∇ + A ) 2 + V + “ P N j =1 γ j δ x j ” , (1.1) where A is a smo oth v ector p oten tial and V a regular electrostatic p oten tial. The p oints x j ∈ R d iden tify the p ositions of the scatterers, although the expression ab o v e is purely formal (see b elow) and the zero- range p erturbation should not b e thought of as a p oten tial. Our aim is to analyze the homogenization (or mean-field) regime in which the num ber of scatterers N diverges, while they cluster within a fixed space region and their in teraction strengths scale prop ortionately . In this limit, the collective effect of the scatterers results in an additional electrostatic p oten tial, determined by the distribution and individual strengths of the centers. This t yp e of problem in three dimensions was first addressed in [FHT98] (see also [FT92, FT93] and references therein), where the authors treated the p oint interactions as indep endent and iden tically dis- tributed random v ariables. Under a suitable scaling, they sho wed that the asso ciated family of Hamil- tonians conv erge in probabilit y , in strong resolven t sense, to a Schr¨ odinger op erator with an additional electrostatic potential, and they also pro vided quan titativ e estimates for the fluctuations around this limit. Their approach relies on explicit resolven t formulas, combined with earlier homogenization results for the Laplacian on domains with many small holes (see, e.g., [R T75, PV80, Oz83, FOT85, F OT87]). F or related w orks in one or more dimensions we refer to [BFT98] and [EN03, Oz06], resp ectively . A somewhat similar mo del is the so-called Ra yleigh gas, which is supp osed to describ e the lo w-energy zero-range in teraction of a quantum particle with N scattering centers [DFT05, CDF08], which are harmonically b ounded to their equilibrium positions. Let us also men tion that the approach originally dev elop ed in [FHT98] was applied more recently in [BCT18] to deriv e a probabilistic characterization of the effective dynamics of a quantum particle in a quantum Lorentz gas with Gross–Pitaevskii-type interactions. Explicit Krein-type resolv en t formulas pla y a crucial role in all the works mentioned abov e. In con trast, here we dev elop an approac h based on quadratic forms and Γ-conv ergence tec hniques, using standard results to deduce strong resolv ent conv ergence as a direct consequence. W e refer to [Br02, DM93] for classical references on Γ-conv ergence. Our framework is arguably more flexible, allowing the inclusion of smo oth electromagnetic bac kground fields and a unified treatmen t of models in dimensions d = 2 and d = 3. The one-dimensional case could also be included, but we omit it since it is qualitatively m uc h simpler (see, e.g., [BFT98]). F urthermore, our assumptions seem to b e more natural, b eing closer in spirit to those in [FHT98], and av oiding s ome of the technical requests in tro duced in [FHT98, EN03, Oz06]. W e also stress that Γ-conv ergence allow us to upgrade the conv ergence to uniform resolven t sense if the original system has discrete sp ectrum, which o ccurs for instance if the external electromagnetic field induces trapping. Ho w ever, as a dra wbac k w e ha ve to restrict to point in teractions with negativ e scattering lengths to ensure equi-co erciv eness of the quadratic forms. The pap er is organized as follo ws. In § 2, after rep orting the main definitions and fixing the notation, w e present the precise assumtions and outline a heuristic deriv ation of the limiting op erator. Therein, we also state our main rigorous results. § 3 collects some preliminaries on Γ-con vergence, complex measures, in tegral kernels and Hilb ert scales. In § 4, we treat separately the pro ofs of the Γ − lim inf and Γ − lim sup, and the pro of of uniform resolven t conv ergence under the trapping assumption. Ac kno wledgments. The present researc h has b een supp orted by the MUR grant “Dipartimen to di Eccellenza 2023-2027” of Dipartimen to di Matematica, P olitecnico di Milano and PRIN 2022 gran t “ONES - Op eN and Effective Quantum Systems” (n. 2022L45W A3). DF ackno wledges the supp ort of INdAM-GNFM through Pr o getto Giovani 2020 “Emergent F eatures in Quantum Bosonic Theories and Semiclassical Analysis”. MC ackno wledges the supp ort of PNRR Italia Domani and Next Generation Eu through the ICSC National R ese ar ch Centr e for High Performanc e Computing, Big Data and Quantum Computing . DC and DF are grateful to Alessandro T eta for insightful and stim ulating discussions. HOMOGENIZA TION OF POINT INTERACTIONS 3 2. Setting and Main Resul ts Let us first consider the Schr¨ odinger op erator H 0 := ( − i ∇ + A ) 2 + V , (2.1) where A ∈ C ∞ ( R d ; R d ) gro ws at most p olynomially at ∞ , and V ∈ L p loc ( R d ; R + ) for some p > 3 is non- negativ e. These assumptions are certainly not optimal and there is room for impro v ement (see Remark 2.1 b elo w), but w e stic k to them for the sak e of concreteness. Note, in particular, that the cases of uniform external magnetic fields and p olynomial trapping electric p otentials are included. W e also remark that H 0 is a self-adjoint op erator on L 2 ( R d ), with domain D ( H 0 ) := n ψ ∈ L 2 ( R d )   ( − i ∇ + A ) 2 ψ , V ψ ∈ L 2 ( R d ) o . It is evident that H 0 is p ositive definite, thus in particular σ ( H 0 ) ⊂ [0 , ∞ ). T aking this in to account, in the sequel we will refer to the asso ciated resolv ent op erator, i.e., for λ > 0, R λ 0 := ( H 0 + λ 2 ) − 1 : L 2 ( R d ) → D ( H 0 ) , and to its integral k ernel (a.k.a. Green function) G λ 0 ( x , y ). Let us also mention that the quadratic form corresp onding to the Hamiltonian H 0 is given by D [ Q 0 ] := n ψ ∈ L 2 ( R d )   ( − i ∇ + A ) ψ , √ V ψ ∈ L 2 ( R d ) o , Q 0 [ ψ ] :=   ( − i ∇ + A ) ψ   2 2 +   √ V ψ   2 2 , where ∥·∥ 2 := ∥·∥ L 2 ( R d ) for short. A prop er self-adjoin t op erator in L 2 ( R d ) matching the heuristic expression (1.1) can b e rigorously defined as a singular p erturbation of H 0 , e.g. , via the construction of its quadratic form, see [T e90]: for any fixed λ > 0, we set D [ Q N ] := n ψ = φ λ + P N j =1 q j G λ 0 ( · , x j ) ∈ L 2 ( R d )   φ λ ∈ D [ Q 0 ] , q 1 , ... , q N ∈ C o , Q N [ ψ ] := Q 0 [ φ λ ] + λ 2 ∥ φ λ ∥ 2 2 − λ 2 ∥ ψ ∥ 2 2 + P N i,j =1 q i  Ξ λ N  ij q j , (2.2) where q i is the complex conjugate of q i and Ξ λ N is the N × N Hermitian matrix with entries  Ξ λ N  ij :=  α j + lim x → x j  G λ 0 0 ( x , x j ) − G λ 0 ( x , x j )   δ ij − G λ 0 ( x i , x j ) (1 − δ ij ) , for i, j ∈ { 1 , ... , N } . (2.3) Here, λ 0 > 0 is a fixed reference parameter and α j ∈ R are finite constan ts labeling the form 1 Q N . W e note that the co efficients α j in tro duced here differ from those used in [AGH+88] by a finite additive constant, dep ending on λ 0 . By direct generalization of the arguments outlined in [T e90], it can b e inferred that the quadratic form Q N is indep endent of λ > 0, closed and b ounded from b elow, for b oth d = 2 and d = 3. Therefore, it uniquely identifies a self-adjoint op erator in L 2 ( R d ). This is giv en b y D ( H N ) = n ψ = φ λ + P N j =1 q j G λ 0 ( · , x j ) ∈ L 2 ( R d )   φ λ ∈ D [ H 0 ] , q 1 , ... , q N ∈ C , and φ λ ( x i ) = P N j =1  Ξ λ N  ij q j for i = 1 , ... , N o inter actio,  H N + λ 2  ψ =  H 0 + λ 2  φ λ . (2.4) 1 F or d = 3 and zero electromagnetic potentials, the constan ts α j and the coefficients γ j in (1.1) satisfy the formal identit y γ j = − c ε + α j ε 2 , for a univ ersal constant c > 0 and for ε → 0, see [BF61] and [AGH+88, § I I.1.2 and App endix H], i.e. , the co efficien ts α j are obtained through a suitable renormalization pro cedure from the “bare” parameters γ j . Similar relations hold for d = 2 and non-zero electromagnetic fields. 4 DOMENICO CAFIERO, MICHELE CORREGGI, AND D A VIDE FERMI The resolv ent R λ N := ( H N + λ 2 ) − 1 : L 2 ( R 2 ) → D ( H N ) is an integral op erator with kernel (see [AGH+88, § I I.1.1, Eq. (1.1.33)] and [P o01]) G λ N ( x , y ) = G λ 0 ( x , y ) + P N i,j =1  Ξ λ N  − 1 ij G λ 0 ( x , x i ) G λ 0 ( x j , y ) . As an aside, let us also men tion that, up on v arying the co efficien ts α j ∈ R ∪ {∞} , the corresp onding family of Hamiltonians H N comprises all the so-called lo c al self-adjoint extensions of the (closable) symmetric op erator H 0 ↾ C ∞ c ( R d \ { x 1 , ... , x N } ). Remark 2.1 (Regularity of A and V ) . The stated h yp otheses on the external p otentials A and V are not meant to b e optimal. The main reason why we assume suc h a regularity on the external p oten tials is to con trol the b ehavior of the Green function G λ 0 ( x , y ) on the diagonal, exploiting results prov en in [BK13, BGP07] (see Lemmas 3.2 and 3.3 b elo w), whic h presumably hold also for less regular A and V . Ho w ever, a throughout analysis of the minimal regularity conditions lies b eyond the scop e of this work. Note also that the p ositivity assumption on V can b e relaxed and b oundedness from b elow is certainly enough (it suffices to shift the whole quadratic form of a p ositive energy), but Kato-smallness of the negativ e part of V is also fine. On the other hand, let us emphasize that the b eha vior of A , V at infinity is unconstrained. In particular, these p oten tials are allow ed to diverge at infinit y , whic h may result in a trapping mechanism for the particle whose dynamics is gov erned by H 0 . Remark 2.2 (W ell-p osedness of Q N and H N ) . W e first notice that, for any fixed y ∈ R d , the function x 7→ G λ 0 ( x , y ) is indeed square-integrable on R d , d ⩽ 3 [BGP07, Thm. 19]. Moreov er, as a consequence of the second resolven t iden tit y , the difference G λ 0 0 ( x , y ) − G λ 0 ( x , y ) is jointly contin uous in ( x , y ) ∈ R d × R d , again if d ⩽ 3 [BGP05, Lemma 9]. This ensures that the limit for x → x j app earing in the definition (2.3) of Ξ λ N exists and is finite. W e stress that this limit do es not dep end on N . On the other hand, in view of the basic identit y G λ 0 ( x , y ) = G λ 0 ( y , x ), it is easy to chec k that the matrix Ξ λ N is Hermitian; in particular, the diagonal limit mentioned abov e is real-v alued. Accordingly , the form Q N is also real-v alued. As an aside, let us mention that when A = 0 (no magnetic field), the Hamiltonian H 0 comm utes with complex conjugation; in this case G λ 0 ( x , y ) is real v alued and th us symmetric, i.e. , G λ 0 ( x , y ) = G λ 0 ( y , x ). F urther prop erties of G λ 0 ( x , y ) will b e discussed in § 3.3. Finally , since D ( H 0 ) ⊂ H 2 loc ( R d ), and hence D ( H 0 ) ⊂ C 0 ( R d ) b y Sob olev em b edding [CP82, Prop. 2.13], the point wise ev aluation φ λ ( x i ) app earing in the definition (2.4) of D ( H N ) is well defined. Our goal in this work is to analyze the limiting b ehavior of H N as N → ∞ in a mean-field regime, where the in teraction centers accumulate within a fixed and b ounded spatial region, while their individual in tensities w eaken prop ortionally , so as to k eep the total strength finite. More precisely , w e shall henceforth refer to the following assumptions ab out the p ositions of the p oints { x j } j ∈ N and the parameters { α j } j ∈ N . Assumption 1 (Distribution of p oints) . There exists a non-negativ e function U ∈ L ∞ ( R d ), with compact support and ∥ U ∥ L 1 = 1, such that x j ∈ supp U for all j ∈ N and 2 µ N := 1 N N X j =1 δ x j w − − − − → N →∞ µ ∞ , with d µ ∞ ( x ) = U ( x ) d x . (2.5) Assumption 2 (Minimal distance) . There exists a constant  > 0, indep enden t on N , such that inf 1 ⩽ i < j ⩽ N | x i − x j | ⩾  N − 1 /d , ∀ N ∈ N . (2.6) 2 Here, we denote by w the w eak conv ergence of measures, i.e. , the conv ergence in the dual of contin uous functions with supp ort con taining supp ( U ) (see § 3.2). HOMOGENIZA TION OF POINT INTERACTIONS 5 supp U x j x i r N Figure 1. A sc hematic illustration of Assumption 1 and Assumption 2. The p oints x j cluster within a fixed region following the densit y distribution U , while maintaining a minimal mutual distance r >  N − 1 /d , see (2.6). Assumption 3 (Interaction strengths) . There exists a b ounded p ositive function a ∈ C 0 ( R d ) such that α j = N a ( x j ) , ∀ N ∈ N and ∀ j ∈ { 1 , ... , N } . (2.7) Remark 2.3 (Assumptions) . Assumption 1 indicates that U serves as the limit density distribution for the p oin ts at whic h the singular p oten tials are concen trated. In particular, the supp ort of U identifies the region of space where the p oints accum ulate. On the other hand, Assumption 2 constrains the rate at whic h the p oints ma y cluster. This condition is crucial to guarantee the equi-coerciveness of the quadratic forms, a k ey element to establish Γ-conv ergence. The latter Assumption 3 prescrib es a sp ecific scaling for the in teraction strengths. Considering that the parameters α j are inv ersely related to the scattering lengths of the individual cen ters [A GH+88, § I.1.4], this assumption actually implies that eac h single-cen ter in teraction v anishes as N → + ∞ . Indeed, it is easy to see that the choice α j = + ∞ , ∀ j , in (2.3) iden tifies the unp erturb ed op erator H 0 . Before stating our main results, let us heuristically describ e the mec hanism of con vergence. The rescaling of the parameters { α j } j ∈{ 1 ,... ,N } as in Assumption 3 implies that the c harges { q j } j ∈{ 1 ,... ,N } also scales with N as q j = 1 N p j , for all j = 1 , ... , N , (2.8) for some complex co efficients p j of order 1. Supp ose no w that there exists a sufficiently regular function p : R 2 → C , which is indep endent of N and suc h that p j = p ( x j ) , for all j = 1 , ... , N . (2.9) Then, by a Riemann sum approximation, the last term in the expression (2.2) b ecomes 1 N 2 N X i,j =1 p i  Ξ λ N  ij p j = 1 N 2 N X i,j =1 p ( x i )  ( N a ( x j ) + O (1)) δ ij − G λ 0 ( x i , x j ) (1 − δ ij )  p ( x j ) = 1 N N X j =1 a ( x j ) | p ( x j ) | 2 − 1 N 2 X i  = j p ( x i ) G λ 0 ( x i , x j ) p ( x j ) + o (1) − − − − → N → ∞ ˆ R d a ( x ) | p ( x ) | 2 U ( x ) d x − ˆ R d × R d p ( x ) G λ 0 ( x , y ) p ( y ) U ( x ) U ( y ) d x d y . 6 DOMENICO CAFIERO, MICHELE CORREGGI, AND D A VIDE FERMI F rom here, assuming that φ N ,λ → φ ∞ ,λ in the appropriate top ology , w e infer that Q N [ ψ N ] + λ 2 ∥ ψ N ∥ 2 2 − − − − → N → ∞ ∥ ( − i ∇ + A ) φ ∞ ,λ ∥ 2 2 +   √ V φ ∞ ,λ   2 2 + λ 2 ∥ φ ∞ ,λ ∥ 2 2 + ⟨ p | a U p ⟩ − D U p    R λ 0 U p E . (2.10) F urthermore, exploiting the b oundary condition (2.4) in the op erator domain D ( H N ), we get φ N ,λ ( x i ) = 1 N N X j =1 h ( N a ( x j ) + O (1)) δ ij − G λ 0 ( x − x j ) (1 − δ ij ) i p ( x j ) − − − − → N →∞ a ( x i ) p ( x i ) − ˆ R d G λ 0 ( x i , y ) p ( y ) U ( y ) d y , whic h ultimately suggests that ψ ∞ ( x ) = a ( x ) p ( x ) . (2.11) Hence, recalling (2.4), ψ N ( x ) = φ N ,λ ( x ) + 1 N N X j =1 p ( x j ) G λ 0 ( x , x j ) − − − − → N → ∞ φ ∞ ,λ + ˆ R d G λ 0 ( x , y ) p ( y ) U ( y ) d y , (2.12) so that  ψ ∞   H 0 + λ 2   ψ ∞  = Q 0 [ φ ∞ ,λ ] + λ 2 ∥ φ ∞ ,λ ∥ 2 2 + 2 ℜ ⟨ φ ∞ ,λ | U p ⟩ + D R λ 0 U p    U p E = Q 0 [ φ ∞ ,λ ] + λ 2 ∥ φ ∞ ,λ ∥ 2 2 + ⟨ p | a U p ⟩ − D U p    R λ 0 U p E +  ψ ∞   U a ψ ∞  , where we hav e exploited the fact that ⟨ ψ ∞ | U p ⟩ =  ψ ∞   U a ψ ∞  = ⟨ p | a U p ⟩ . Summing up,  ψ ∞   H 0 − U a   ψ ∞  + λ 2 ∥ ψ ∞ ∥ 2 2 = Q 0 [ φ λ, ∞ ] + λ 2 ∥ φ ∞ ,λ ∥ 2 2 + ⟨ p | a U p ⟩ − D U p    R λ 0 U p E , (2.13) whic h yields the exp ession of the limiting quadratic form b y comparison of (2.10) with (2.13): Q ∞ [ ψ ∞ ] = Q 0 [ ψ ∞ ] −  ψ ∞   U a   ψ ∞  , (2.14) where U and a are the functions introduced previously in Assumption 1 and Assumption 2. Note that, under these assumptions, the ratio U /a b elong to L ∞ ( R d ), which ensures that, if the guess if correct, Q ∞ is a Kato-small p erturbation of the initial quadratic form Q 0 and D [ Q ∞ ] = D [ Q 0 ]. F urthermore, Q ∞ is closed and b ounded from b elow, and therefore it uniquely iden tifies a self-adjoin t op erator, namely , H ∞ = H 0 − U a , D ( H ∞ ) = D ( H 0 ) . (2.15) W e are now ready to state our main result. Here and in the sequel, we understand that the quadratic forms Q N and Q ∞ are extended to the entire Hilb ert space L 2 ( R d ) by setting them equal to + ∞ outside of their resp ective domains of definition. Theorem 2.1 (Γ − conv ergence) . L et Assumptions 1 to 3 hold. Then, the se quenc e of quadr atic forms { Q N } N ∈ N Γ -c onver ges to Q ∞ with r esp e ct to b oth the we ak and str ong top olo gy of L 2 ( R d ) . Mor e pr e cisely, the fol lowing two c onditions ar e fulfil le d: (i) (Γ − lim inf ) for any se quenc e { ψ N } N ∈ N ⊂ L 2 ( R d ) such that ψ N w − − − − − → N → + ∞ ψ ∞ ∈ L 2 ( R d ) , ther e holds Q ∞ [ ψ ∞ ] ⩽ lim inf N → + ∞ Q N [ ψ N ] ; (2.16) HOMOGENIZA TION OF POINT INTERACTIONS 7 (ii) (Γ − lim sup) for any ψ ∞ ∈ D [ Q ∞ ] , ther e exists a se quenc e { ψ N } N ∈ N ⊂ L 2 ( R d ) , with ψ N ∈ D [ Q N ] for al l N ∈ N , such that ψ N − − − − − → N → + ∞ ψ ∞ ∈ L 2 ( R d ) and Q ∞ [ ψ ∞ ] ⩾ lim sup N → + ∞ Q N [ ψ N ] . (2.17) As a direct consequence of Theorem 2.1, by standard arguments of Γ-con v ergence theory , w e obtain the follo wing result regarding the family of self-adjoint op erators { H N } N ∈ N and the limit Hamiltonian H ∞ . Corollary 2.1 (Op erator conv ergence) . L et Assumptions 1 to 3 hold. Then, the se quenc e { H N } N ∈ N c onver ges to H ∞ in str ong r esolvent sense as N → ∞ . Mor e over, if H 0 has c omp act r esolvent, then { H N } N ∈ N c onver ges to H ∞ in norm r esolvent sense. Remark 2.4 (Bound states and dynamics) . Since U and a are b oth non-negativ e, the limiting Hamiltonian H ∞ comprises an attractive electrostatic p otential which ma y supp ort negativ e-energy b ound states. It is worth noting that also the Hamiltonians H N t ypically p ossess a non-empt y discrete sp ectrum b elo w the con tin uous threshold [A GH+88, § I I.1.1, Thm. 1.1.4]. The result of Γ-conv ergence established in Theorem 2.1 guaran tees any sequence of ground states of { H N } N ∈ N con v erges to the ground state of H ∞ [DM93, Chpt. 7]. On top of that, when H 0 has compact resolv en t, the uniform resolven t conv ergence stated in Corollary 2.1 implies that the purely discrete sp ectrum of H 0 actually consists of eigen v alues whic h arise as accumulation p oints of sequences of eigenv alues of H N [DeO09, Prop. 10.2.4 and Cor. 10.2.5]. Let us finally recall that strong resolv en t con v ergence is in fact equiv alen t to strong conv ergence of the unitary op erators e − itH N to e − itH ∞ , for an y t ∈ R , and thus it also provides the conv ergence of the corresp onding quan tum dynamics [DeO09, Prop. 10.1.8]. Remark 2.5 (Relaxing the assumptions) . Let us briefly comment on possible relaxations of Assumptions 1 to 3. On the one hand, the theory developed here is stable under mild violations of these hypotheses. The same conclusions w ould hold if a small n um ber of in teractions, sa y of order O (1) as N → ∞ , did not satisfy the stated assumptions. Likewise, the compact-supp ort requiremen t on the density U could b e replaced b y a sufficien tly fast deca y at infinit y , at the price of some additional effort. Subleading corrections in Assumption 3 could also b e incorp orated; for instance, one may allow α j = N a ( x j ) + O (1) as N → ∞ . On the other side, it would p ose ma jor challenges to consider a large num ber of non-p ositive interaction parameters α j ⩽ 0 or to p ermit the centers to accumulate faster than allow ed by the low er b ound (2.6) on the mutual distances. In these cases, the equi-co erciv eness of the quadratic forms Q N w ould fail. As a result, the b eha vior of the system could b ecome significantly different, p otentially exhibiting new phenomena that are not captured by the curren t analysis. Remark 2.6 (Domains with b oundaries and curved geometries) . Our arguments could b e extend with minor effort to flat spatial domains with b oundaries. More precisely , the pro ofs remain essentially un- c hanged if, instead of working on R d , one w ere to consider a domain Ω, b ounded or un b ounded, with a sufficien tly regular b oundary ∂ Ω. The only structural requirement is that the differential op erator H 0 ad- mits a self-adjoin t realization on L 2 (Ω), whic h in turn relies on specifying appropriate boundary conditions on ∂ Ω. T o a v oid tec hnical complications, one should p erhaps ensure that the p oints where the singular p oten tials are supp orted do not accumulate near the b oundary . In the same spirit, one could even extend our results to analogous mo dels living on curved Riemannian manifolds, with the obvious adjustments to the definitions of the basic quadratic forms and related Hamiltonian op erators. 3. Preliminaries In this section, w e recall some basic facts on Γ-con v ergence, complex measures, integral kernels and Hilb ert scales, whic h will b e used in the pro ofs to b e presented in the next section. F or more details on these topics w e refer to [Br02, DM93], [Bo18, F o99, Ru87], [BGP05, BGP07] and [AKN07, F e16], resp ectiv ely (see also [FP17, App. B] and [FP23]). 8 DOMENICO CAFIERO, MICHELE CORREGGI, AND D A VIDE FERMI 3.1. Γ − con v ergence. Consider a sequence { F N } N ∈ N of closed and low er-bounded quadratic forms on a Hilb ert space H with domains D [ F N ]. Let F ∞ on D [ F ∞ ] b e another low er-bounded quadratic form on H , to b e regarded as the candidate limit of the sequence. W e denote by A N ( N ∈ N ) and A ∞ the self-adjoin t op erators asso ciated with F N and F ∞ , resp ectively . In the sequel, the forms F N ( N ∈ N ) and F ∞ are implicitly extended (relaxed) to the whole Hilb ert space H by setting their v alues equal to + ∞ outside of their resp ective domains. Let τ denote either the strong or weak top ology on H . The sequence { F N } N ∈ N is said to Γ -c onver ge to F ∞ w.r.t. τ on H if the following tw o conditions are fulfilled: (i) (Γ − lim inf ) for an y sequence { ψ N } N ∈ N ⊂ H such that ψ N τ − → ψ ∞ ∈ H , there holds F ∞ [ ψ ∞ ] ⩽ lim inf N → ∞ F N [ ψ N ] ; (ii) (Γ − lim sup) for an y ψ ∞ ∈ D [ F ∞ ], there exists a sequence { ψ N } N ∈ N ⊂ D [ F N ], such that ψ N τ − → ψ ∞ and F ∞ [ ψ ∞ ] ⩾ lim sup N → ∞ F N [ ψ N ] . Since F ∞ is low er semicontin uous and b ounded from b elow, without loss of generality we may replace condition (ii) with the following weak er requirement, see [Br02, Rem. 1.29] and [DFT94, Lemma 4.2(ii)]: (ii’) for any ψ ∞ ∈ D ( A ∞ ), there exists a sequence { ψ N } N ∈ N ⊂ D [ F N ], such that ψ N τ − → ψ ∞ and F ∞ [ ψ ∞ ] = lim N →∞ F N [ ψ N ] . (3.1) The main result relating Γ-conv ergence of quadratic forms and strong conv ergence of the corresp onding resolv en t op erators is the follo wing (see, e.g. , [DM93, Thm. 13.6] and [BdO+14, Thm. 1]): Theorem 3.1 (Γ − cov ergence and op erator conv ergence 1) . Assume the quadr atic forms { F N } N ∈ N and F ∞ to b e lower semic ontinuous and uniformly b ounde d fr om b elow. Then, the fol lowing statements ar e e quivalent: (i) { F N } N ∈ N Γ -c onver ges to F ∞ w.r.t. the str ong top olo gy and c ondition (i) is valid w.r.t. the we ak top olo gy; (ii)  F N + λ 2 ∥ · ∥ 2  N ∈ N Γ -c onver ges to F ∞ + λ 2 ∥ · ∥ 2 w.r.t. b oth the str ong and we ak top olo gy, for some λ ⩾ 0 ; (iii) { A N } N ∈ N c onver ges to A ∞ in the str ong r esolvent sense. T o relate Γ − conv ergence and uniform resolv en t conv ergence, it is necessary to in tro duce a notion of compactness [Mo94]. A sequence of quadratic forms { F N } N ∈ N , uniformly b ounded from b elo w b y λ ∗ ∈ R , is called asymptotic al ly c omp act in H , if for any sequence { ψ N } N ∈ N ⊂ H such that lim inf N → ∞  F N [ ψ N ] + λ 2 ∗ ∥ ψ N ∥ 2  < ∞ , (3.2) there exists a subsequence { ψ N n } n ∈ N , which con verges strongly in H . By a simple v ariation of [DeO11, Prop. 4] (see also [KS03]), we obtain the following: Theorem 3.2 (Γ − cov ergence and op erator conv ergence 2) . Assume the quadr atic forms { F N } N ∈ N and F ∞ to b e lower semic ontinuous and uniformly b ounde d fr om b elow. Mor e over, assume that (i) { F N } N ∈ N Γ − c onver ges to F w.r.t. b oth the str ong and we ak top olo gy; (ii) { F N } N ∈ N is asymptotic al ly c omp act in H ; (iii) ( A ∞ + λ 2 ) − 1 is c omp act for some λ > 0 lar ge enough; then, { A N } N ∈ N c onver ges to A ∞ in norm r esolvent sense. HOMOGENIZA TION OF POINT INTERACTIONS 9 3.2. Measures and distribution of cen ters. W e discuss here some implications of Assumptions 1 and 2 for complex measures related to the empiric measure µ N in tro duced in (2.5). T o this purp ose we first briefly recall some basic facts on complex measures to fix the notation. F or any complex measure µ on the standard Borel σ − algebra B ( R d ) of R d , its v ariation | µ | is given b y | µ | ( E ) = sup ∞ X n =1 | µ ( E n ) | , for all E ∈ B ( R d ) , where the suprem um is taken ov er all partitions { E n } ∞ n =1 of E . The total v ariation ∥ µ ∥ TV := | µ | ( R d ) defines a norm on the space of complex measures M C ( R d ), which mak es the latter a Banach space [F o99, Prop. 7.16]. A sequence of (complex) measures { µ N } N ∈ N ⊂ M C ( R d ) con verges in the weak sense to µ ∈ M C ( R d ) (denoted as µ N w − − − − − → N → + ∞ µ ), if lim N →∞ ˆ R d f d µ N = ˆ R d f d µ, for all f ∈ C b ( R d ) , (3.3) where C b ( R d ) is the space of all contin uous and b ounded complex-v alued functions on R d . Finally , we recall that a sequence of measures { µ N } N ∈ N ⊂ M C ( R d ) is uniformly tight if ∀ ε > 0 there exists a compact set K ε ⊂ R d suc h that | µ N | ( R d \ K ε ) ⩽ ε for all N ∈ N . Lemma 3.1. L et Assumptions 1 and 2 hold and let { ν N } N ∈ N b e the se quenc e of c omplex me asur es ν N = 1 N N X j =1 p j δ x j , (3.4) wher e p j ∈ C ar e so that ther e exists a c onstant c > 0 indep endent of N such that 1 N N X j =1 | p j | 2 ⩽ c . (3.5) Then, ther e exists a we akly c onver gent subse quenc e { ν N k } k ∈ N , with ν N k w − → ν ∞ ∈ M C ( R d ) . Mor e over, the limit ν ∞ is absolutely c ontinuous with r esp e ct to µ ∞ , i.e., ther e exists p ∈ L 2 ( R d , d µ ∞ ) such that d ν ∞ = p d µ ∞ . (3.6) Pr o of. Let us firstly notice that Assumption 1 ensures the uniform tightness of the sequence { ν N } N ∈ N . F urthermore, by Cauc hy-Sc hw artz inequality and condition (3.5), it follows that ∥ ν N ∥ 2 TV ⩽  1 N P N j =1 | p j |  2 ⩽ 1 N P N j =1 | p j | 2 ⩽ c , whic h shows that the sequence { ν N } N ∈ N is also uniformly b ounded in total v ariation. Then, by Prokhoro v theorem [Bo18, Thm. 1.4.11], w e infer the existence of a w eakly con v ergent subsequence { ν N k } k ∈ N , with ν N k w − − − − → k → + ∞ ν ∞ ∈ M C ( R d ). No w, consider the linear functionals on C b ( R d ) defined by µ N k ( f ) = ˆ R d f d µ N k , ν N k ( f ) = ˆ R d f d ν N k , µ ∞ ( f ) = ˆ R d f d µ ∞ , ν ∞ ( f ) = ˆ R d f d ν ∞ . Again by Cauch y-Sc h w artz inequality , for an y k ∈ N and for all f ∈ C b ( R d ), | ν N k ( f ) | 2 =    1 N k P N k j =1 p j f ( x j )    2 ⩽  1 N k P N k j =1 | p j | 2   1 N k P N k j =1 | f ( x j ) | 2  ⩽ c ˆ R d | f | 2 d µ N k = c µ N k  | f | 2  . 10 DOMENICO CAFIERO, MICHELE CORREGGI, AND D A VIDE FERMI On the other hand, by weak conv ergence of { ν N k } k ∈ N and { µ N k } k ∈ N , we deduce that | ν N k ( f ) | → | ν ∞ ( f ) | and | µ N k ( | f | 2 ) | → | µ ∞ ( | f | 2 ) | , as k → ∞ , since | f | 2 ∈ C b ( R d ). So, w e can pass to the limit and obtain | ν ∞ ( f ) | 2 ⩽ c µ ∞  | f | 2  = c ∥ f ∥ 2 L 2 ( R d , d µ ∞ ) . By standard density arguments, this implies that ν ∞ extends to a b ounded linear functional on L 2 ( R d , d µ ∞ ). Therefore, by Riesz theorem [F o99, Thm. 7.17] (see also [Ru87, Thm. 6.19]), we infer the existence of a function p ∈ L 2 ( R d , d µ ∞ ) such that ν ∞ ( f ) = ˆ R d f p d µ ∞ , for all f ∈ L 2 ( R d , d µ ∞ ) . Finally , since U has compact supp ort by Assumption 1, we also hav e p ∈ L 1 ( R d , d µ ∞ ), which pro ves that ν ∞ is absolutely contin uous w.r.t. µ ∞ . □ Under the assumptions of Lemma 3.1, b y linearity , the sequence of complex conjugate measures { ν N } N ∈ N giv en b y ν N = 1 N N X j =1 p j δ x j , admits a subsequence which weakly con v erges to ν ∞ . Of course, the latter fulfills d ν ∞ = p d µ U , where p ∈ L 2 ( R d , d µ ∞ ) is the function identified in (3.6). 3.3. Green functions. Consider a Schr¨ odinger op erator of the form (2.1) and let G λ 0 ( x , y ), with λ > 0 and ( x , y ) ∈ R d × R d , b e the in tegral kernel asso ciated to its resolv en t. Some key prop erties of G λ 0 ( x , y ) w ere already discussed in Remark 2.2. Here we rep ort some additional information on p oin twise b ounds and on the singular b ehavior of G λ 0 ( x , y ) near the diagonal { x = y } . Standard results on Carleman k ernels ensure that G λ 0 ( x , y ) is join tly con tinuous a w a y from the diagonal [BGP07, Thm. 21]: G λ 0 ∈ C 0 ( R d × R d \ { x = y } ) . F uther prop erties of G λ 0 ( x , y ) are conv eniently formulated by comparison with the (real-v alued) Green function g λ 0 ( x − y ) for the free Laplacian: g λ 0 ( x ) = ( 1 2 π K 0  λ | x |  , if d = 2 , e − λ | x | 4 π | x | , if d = 3 . (3.7) Here K 0 denotes the mo dified Bessel function of the second kind and order zero ( a.k.a. Macdonald function) [OLB+10, Ch. 10]. Exploiting the p ositivity of the electrostatic p otential V and the diamagnetic inequality , one infers the follo wing p oin twise estimate [BGP07, Lemma 15]. Lemma 3.2. F or any λ > 0 ther e holds | G λ 0 ( x , y ) | ⩽ g λ 0 ( x − y ) for al l ( x , y ) ∈ R d × R d \ { x = y } . (3.8) On the other hand, G λ 0 ( x , y ) shares the same leading order singularit y as g λ 0 ( x − y ) for | x − y | → 0. A more refined asymptotic expansion can b e derived using the standard Laplace-transform representation G λ 0 ( x , y ) = ˆ ∞ 0 e − tλ 2 K 0 ( t ; x , y ) d t , where K 0 ( t ; x , y ) is the in tegral kernel asso ciated to the heat op erator e − tH 0 . Com bining this with the small-time asymptotics of K 0 ( t ; x , y ) for t → 0 + [BK13], one infers the following: HOMOGENIZA TION OF POINT INTERACTIONS 11 Lemma 3.3. F or any fixe d λ > 0 ther e holds G λ 0 ( x , y ) = ζ 0 ( x − y ) e iϑ A ( x , y ) + h λ ( x , y ) , (3.9) wher e ζ 0 ( x ) := ( − 1 2 π log | x | , if d = 2 , 1 4 π | x | , if d = 3 , (3.10) ϑ A ( x , y ) := ( x − y ) · ˆ 1 0 A  y + s ( x − y )  d s . (3.11) and h λ ( x , y ) is a jointly c ontinuous function on R d × R d . Remark 3.1 (Lo cal singularity of G λ 0 ( x , y )) . The term ζ 0 ( x − y ) in (3.9) captures a universal singular b eha vior on the diagonal. The magnetic phase ϑ A ∈ C ∞ ( R d × R d ) is essential when d = 3 and A  = 0 . In fact, it accounts for a subleading contribution prop ortional to ( x − y ) / | x − y | in the asymptotic expansion, whic h would otherwise sp oil the contin uit y of the reminder h λ ( x , y ) across the diagonal. In all other cases ( i.e. , d = 2 for an y A , or d = 3 with A = 0 ), the reminder is con tinuous regardless of the phase correction. 3.4. Hilb ert scales asso ciated to H 0 . Exploiting the p ositivity and self-adjoin tness of H 0 , w e can define the fractional pow ers ( H 0 + 1 ) r/ 2 for any r ∈ R , and, equipping the domain D  ( H 0 + 1 ) r/ 2  with the inner pro duct ⟨ ψ 1 | ψ 2 ⟩ r := D ( H 0 + 1 ) r/ 2 ψ 1    ( H 0 + 1 ) r/ 2 ψ 2 E , (3.12) w e can construct the scale of Hilb ert spaces H r := D  ( H 0 + 1 ) r/ 2  ∥·∥ H r , for r ∈ R , where ∥ · ∥ H r indicates the norm induced b y the mo dified inner product (3.12). Then, w e ha v e the follo wing prop ert y [F e16, Prop. 2.16 and Prop. 2.18]. Lemma 3.4. F or any r > 0 , H − r = ( H r ) ′ . Next, combining lo cal elliptic regularity for p ositiv e even orders with complex interpolation theory and standard Sob olev embedding theorem [CP82, p. 98, Prop. 2.13], one obtains the following regularity prop erties [F e16, Prop. 2.42 and Cor. 2.43] (see also [FP23]). Lemma 3.5. The fol lowing inclusions ar e c ontinuous emb e ddings: (i) H r ⊂ H r loc ( R d ) for every r ⩾ 0 ; (ii) H r ⊂ C 0 ( R d ) for every r > d/ 2 . W e conclude by recalling the following result [F e16, Prop. 2.12]. Lemma 3.6. F or any λ > 0 , the r esolvent R λ 0 c an b e uniquely extende d to a b ounde d op er ator R λ 0 : H r → H r +2 , for any r > 0 . (3.13) 4. Proofs F or later conv enience let us mo dify the domain expression (2.2) of the quadratic forms Q N according to (2.8) and use the charges { p j = N q j } j ∈{ 1 ,...,N } instead of { q j } j ∈{ 1 ,...,N } , i.e. , D [ Q N ] := n ψ = φ λ + 1 N P N j =1 p j G λ 0 ( · , x j ) ∈ L 2 ( R d )   φ λ ∈ D [ Q 0 ] , p 1 , ... , p N ∈ C o , Q N [ ψ ] := Q 0 [ φ λ ] + λ 2 ∥ φ λ ∥ 2 2 − λ 2 ∥ ψ ∥ 2 2 + 1 N 2 P N i,j =1 p i  Ξ λ N  ij p j . 12 DOMENICO CAFIERO, MICHELE CORREGGI, AND D A VIDE FERMI Note that the op erator domain (2.4) may b e c hanged accordingly . Let also Q ∞ b e the alleged Γ-limit of Q N , as defined in (2.14). In view of Theorem 3.1, it is enough to pro v e the analogue of Theorem 2.1 for the shifted quadratic forms Q λ N := Q N + λ 2 ∥ · ∥ 2 2 , D [ Q λ N ] = D [ Q N ] , (4.1) Q λ ∞ := Q ∞ + λ 2 ∥ · ∥ 2 2 , D [ Q λ ∞ ] = D [ Q ∞ ] , (4.2) where λ > 0 is a fixed spectral parameter, independent of N , to b e c hosen large enough. As b efore, w e extend these forms to the whole Hilb ert space L 2 ( R d ) by setting them equal to + ∞ outside of their resp ectiv e domains. In the sequel, we establish the Γ − lim inf and Γ − lim sup inequalities for the sequence { Q λ N } N ∈ N , which corresp ond to items (i) and (ii) in Theorem 2.1. These ultimately accoun t for the Γ-con v ergence result rep orted in Theorem 2.1. W e will finally derive Corollary 2.1 exploiting asymptotic compactness and Theorem 3.2. 4.1. Pro of of the Γ − lim inf inequalit y. In this section, we establish item (i) in Theorem 2.1. T o this a v ail, we b egin by collecting a few preliminary auxiliary lemmas. The following result provides a generalization of some arguments, originally presented in [PV80]. Lemma 4.1. Supp ose Assumptions 1 and 2 hold and c onsider the se quenc e of c omplex me asur es { ν N } N ∈ N (3.4) , fulfil ling (3.5) and c onver ging we akly to ν ∞ ∈ M C ( R d ) . Mor e over, let ζ 0 b e define d as in (3.10) and ξ b e a Lipschitz c ontinuous function on R d × R d . Then, lim N →∞ ˆ ( R d × R d ) \ ∆ ζ 0 ( x − y ) ξ ( x , y ) d ν N ( x ) dν N ( y ) = ˆ R d × R d ζ 0 ( x − y ) ξ ( x , y ) d ν ∞ ( x ) d ν ∞ ( y ) , (4.3) wher e ∆ =  ( x , y ) ∈ R d × R d   x = y  . Pr o of. F or the sak e of clarit y , w e first pro v e the result with ξ = 1. W e will comment on the minor adjustmen ts needed to handle a generic Lipschitz con tinuous function ξ at the end of this pro of. F or eac h j ∈ { 1 , ... , N } , let B N ,j b e the op en ball of center y j and radius r N = ℓ 2 N − 1 /d (recall (2.6)). Due to Assumption 2, for any i  = j , w e hav e B N ,i ∩ B N ,j = ∅ and the function y 7→ ζ 0 ( y i − y ) is harmonic on B N ,j . Applying the mean v alue prop erty twice, we obtain ζ 0 ( y i − y j ) = 1 | B N ,i | | B N ,j | ˆ B N ,i × B N ,j ζ 0 ( x − y ) d x d y , for all i  = j . Using the explicit expression (3.4) for ν N and the ab ov e identit y , w e get ˆ ( R d × R d ) \ ∆ ζ 0 ( x − y ) d ν N ( x ) d ν N ( y ) = 1 N 2 N X i,j =1 , i  = j ζ 0 ( x i − x j ) p i p j = 1 N 2 N X i,j =1 , i  = j p i p j | B N ,i | | B N ,j | ˆ B N ,i × B N ,j ζ 0 ( x − y ) d x d y = ˆ ( R d × R d ) \ ∆ ζ 0 ( x − y ) η N ( x ) η N ( y ) d x d y , (4.4) where η N := 1 N N X j =1 p j 1 B N ,j | B N ,j | . It can b e chec ked b y a straigh tforw ard computation that η N d x = d( ν N ∗ ω N ), where ω N is the uniform measure on the ball with center at the origin and radius r N . By assumption and direct insp ection, ν N w − − − − − → N → + ∞ ν ∞ , ω N w − − − − − → N → + ∞ δ 0 . HOMOGENIZA TION OF POINT INTERACTIONS 13 Then, since the conv olution of tw o w eakly conv ergen t sequences of measures also w eakly conv erges to the con v olution of the limit measures [Bo18, Ex. 4.8.49], w e get that ν N ∗ ω N w − → ν ∞ ∗ δ 0 = ν ∞ , or, equiv alently , lim N →∞ ˆ ( R d × R d ) \ ∆ f ( x , y ) d ν N ( x ) d ν N ( y ) = ˆ R d × R d f ( x , y ) d ν ∞ ( x ) d ν ∞ ( y ) , ∀ f ∈ C b ( R d × R d ) . (4.5) Next, w e introduce a contin uous and b ounded regularization of the singular function ζ 0 . Let us first discuss the three-dimensional case. Setting, for ε > 0, ζ ε 0 ( x ) := 1 4 π ( | x | + ε ) , w e ha v e ˆ ( R 3 × R 3 ) \ ∆ ζ 0 ( x − y ) d ν N ( x ) d ν N ( y ) = ˆ ( R 3 × R 3 ) \ ∆ ζ ε 0 ( x − y ) d ν N ( x ) d ν N ( y ) + ˆ ( R 3 × R 3 ) \ ∆ [ ζ ε 0 ( x − y ) − ζ 0 ( x − y )] d ν N ( x ) d ν N ( y ) . (4.6) Ho w ever, since ζ ε 0 ( x − y ) ∈ C b ( R 3 × R 3 ), by (4.5), lim N →∞ ˆ R 3 × R 3 \ ∆ ζ ε 0 ( x − y ) d ν N ( x ) d ν N ( y ) = ˆ R 3 × R 3 ζ ε 0 ( x − y ) d ν ∞ ( x ) d ν ∞ ( y ) . (4.7) On the other hand, the sequence { η N } N ∈ N is uniformly b ounded in L 2 ( R 3 ): recalling (3.5), ∥ η N ∥ 2 2 = 1 N 2 N X j =1 | p j | 2 | B N ,j | = 6 π  3 1 N N X j =1 | p j | 2 ⩽ 6 c π  3 . F urthermore, by Assumption 1, supp ( η N ) is con tained inside a fixed b ounded domain indep endent of N , whic h implies that the sequence { η N } N ∈ N is uniformly b ounded in L p ( R 3 ), for an y 1 ⩽ p ⩽ 2. Then, b y using the Hardy-Littlewoo d-Sob olev inequality [LL01, Thm. 4.3],      ˆ ( R 3 × R 3 ) \ ∆ [ ζ ε 0 ( x − y ) − ζ 0 ( x − y )] d ν N ( x ) d ν N ( y )      ⩽ ε 4 π ˆ R 3 × R 3 1 | x − y | 2 | η N ( x ) | | η N ( y ) | d x d y ⩽ C ε ∥ η N ∥ 2 L 2 / 3 ( R 3 ) ⩽ C ′ ε − − − − → ε → 0 + 0 , where C , C ′ > 0 are suitable constants indep enden t of N and ε . It can b e shown in a similar fashion that lim ε → 0 + ˆ ( R 3 × R 3 ) \ ∆ [ ζ ε 0 ( x − y ) − ζ 0 ( x − y )] dν ∞ ( x ) dν ∞ ( y ) = 0 . (4.8) Com bining (4.6)-(4.8) ultimately prov es the thesis (4.3) for ξ = 1 and d = 3. The analogous result with ξ = 1 and d = 2 can b e deriv ed retracing the same arguments, considering the regularized version of g defined as ζ ε 0 ( x ) := − 1 2 π log( | x | + ε ) ( ε > 0) . Strictly sp eaking, such a function is con tinuous but not b ounded, as it diverges at infinity . Ho w ev er, this p oses no real issue, since all the measures inv olved hav e supp orts contained within a fixed compact set. W e conclude b y outlining the mo difications required for a generic Lipsch itz contin uous function ξ . In this case, a straightforw ard telescopic argument sho ws that the analogue of (4.4) b ecomes ˆ ( R d × R d ) \ ∆ ζ 0 ( x − y ) ξ ( x , y ) d ν N ( x ) d ν N ( y ) = ˆ ( R d × R d ) \ ∆ ζ 0 ( x − y ) ξ ( x , y ) η N ( x ) η N ( y ) d x d y + Θ N , 14 DOMENICO CAFIERO, MICHELE CORREGGI, AND D A VIDE FERMI where Θ N := 1 N 2 N X i,j =1 , i  = j p i p j | B N ,i | | B N ,j | ˆ B N ,i × B N ,j ζ 0 ( x − y ) [ ξ ( x i , x j ) − ξ ( x , y )] d x d y . By arguments similar to those outlined in the first part of the pro of, we readily obtain lim N →∞ ˆ ( R d × R d ) \ ∆ ζ 0 ( x − y ) ξ ( x , y ) η N ( x ) η N ( y ) d x d y = ˆ R 3 × R 3 ζ 0 ( x − y ) ξ ( x , y ) d ν ∞ ( x ) d ν ∞ ( y ) . Concerning the reminder Θ N , using the Lipsc hitz contin uit y of ξ and the fact that | x i − x | , | x j − y | ⩽ ℓ 2 N − 1 /d for x ∈ B N ,i and y ∈ B N ,j , we hav e | Θ N | ⩽  √ 2 N 1 /d ∥ ξ ∥ C 0 , 1 N 2 N X i,j =1 , i  = j | p i | | p j | | B N ,i | | B N ,j | ˆ B N ,i × B N ,j ζ 0 ( x − y ) d x d y ⩽ C N 1 /d , for some constant C > 0, which shows that Θ N → 0 as N → ∞ and thus concludes the pro of. □ Let us now pass to discuss the p ositivity of the matrix Ξ λ N . T o this purp ose, let us first rep ort a result pro v en in [Se15, § 2.4, pp. 25-27 and Remark 2.19]. Lemma 4.2. L et Assumptions 1 and 2 hold. Then, for any fixe d s ∈ (0 , d ) , ther e exists a c onstant c s > 0 such that lim sup N →∞ 1 N 2 N X i,j =1 i  = j 1 | x i − x j | s ⩽ ˆ R d × R d 1 | x − y | s d µ ∞ ( x ) d µ ∞ ( y ) ⩽ c s . (4.9) The argumen ts presented therein actually establish that the inequalities in (4.9) remain v alid when the function 1 / | x | s is replaced b y any g ∈ L 1 loc ( R d ) ∩ C 0 ( R d \ { 0 } ) which is monotone radial and p ositive near the origin. Lemma 4.3. L et Assumptions 1 to 3 hold. Then, for any fixe d λ > 0 lar ge enough, ther e exist N λ > 0 and a c onstant γ λ > 0 , indep endent of N , such that 1 N Ξ λ N ⩾ γ λ 1 , for al l N ⩾ N λ . (4.10) Pr o of. Let us recall the definition (2.3) of Ξ λ N and set Ξ λ N = N ( A λ N + B λ N ), where  A λ N  ij := 1 N  α j + lim x → x j  G λ 0 0 ( x , x j ) − G λ 0 ( x , x j )   δ ij ,  B λ N  ij := − 1 N G λ 0 ( x i , x j ) (1 − δ ij ) . (4.11) T aking into accoun t Assumption 3 and the fact that the difference G λ 0 0 ( x , y ) − G λ 0 ( x , y ) is join tly con tin uous on R d × R d for all λ, λ 0 > 0 (see Remark 2.2), it is easy to see that A λ N ⩾ (min a ) 1 + o (1) , as N → ∞ . Then, the thesis follows as so on as w e can establish that the op erator norm of B λ N can b e made arbitrarily small, uniformly with resp ect to N , by pic king λ sufficiently large. T o this purp ose, it is con v enient to consider the Hilb ert-Schmidt norm, noting that    B λ N    2 ⩽    B λ N    2 HS = 1 N 2 X i,j =1 i  = j   G λ 0 ( x i , x j )   2 ⩽ 1 N 2 X i,j =1 i  = j    g λ 0 ( x i − x j )    2 , where the last inequality follows from Lemma 3.2. Let us also recall the explicit expression (3.7) for the free Green function g λ 0 and pro ceed to discuss separately the cases d = 3 and d = 2. HOMOGENIZA TION OF POINT INTERACTIONS 15 F or d = 3, using the elementary inequalit y e − t ⩽ t − β , for t > 0 and β ∈ (0 , 1), together with Lemma 4.2 for s = 2 + β , we obtain   B λ N   2 ⩽ 1 N 2 X i,j =1 i  = j      e − λ | x i − x j | 4 π | x i − x j |      2 ⩽ (2 λ ) − β 16 π 2 N 2 X i,j =1 i  = j 1 | x i − x j | 2+ β ⩽ C λ − β . F or d = 2, using a w ell-kno wn in tegral represen tation of the Bessel function K 0 [OLB+10, Eq. 10.32.10], alongside with the inequality e − t ⩽ t − β and Lemma 4.2, now with s = 4 β and β ∈ (0 , 1 / 2), we get   B λ N   2 ⩽ 1 16 π 2 N 2 X i,j =1 i  = j   K 0  λ | x i − x j |    2 = 1 16 π 2 N 2 X i,j =1 i  = j     ˆ ∞ 0 exp  − t − λ 2 | x i − x j | 2 4 t  d t t     2 = λ − 4 β 2 4(1 − β ) π 2  ˆ ∞ 0 t β − 1 e − t dt  2 1 N 2 X i,j =1 i  = j 1 | x i − x j | 4 β ⩽ C λ − 4 β . Cho osing λ > 0 large enough, the ab ov e arguments imply (4.10) for b oth d = 3 and d = 2. □ W e are now ready to prov e the limit inferior inequality , namely , claim (i) in Theorem 2.1. Prop osition 4.1 (Γ − lim inf ) . L et Assumptions 1 to 3 hold and let λ > 0 b e fixe d lar ge enough. Then, for any se quenc e { ψ N } N ∈ N ⊂ L 2 ( R d ) such that ψ N w − − − − − → N → + ∞ ψ ∞ , ther e holds Q λ ∞ [ ψ ∞ ] ⩽ lim inf N → ∞ Q λ N [ ψ N ] . (4.12) Pr o of. The statement is trivial if the limit inferior on the right-hand side of (4.12) is infinite. Hence, we assume that ψ N ∈ D [ Q N ]. In fact, p ossibly passing to a subsequence, we can also assume that lim inf N → ∞ Q λ N [ ψ N ] = lim N → ∞ Q λ N [ ψ N ] ⩽ c , (4.13) for some finite constant c > 0 indep endent of N . T o av oid a to o hea vy notation how ev er w e still lab el the subsequence by N ∈ N . In view of (2.2), (4.1) and Lemma 4.3, it app ears that each term in Q λ N [ ψ N ] is p ositiv e definite for any sufficien tly large N . Then, the uniform b oundedness of Q λ N [ ψ N ] yields Q 0 [ φ N ,λ ] + λ 2 ∥ φ N ,λ ∥ 2 2 ⩽ c , 1 N 2 P N i,j =1 p i  Ξ λ N  ij p j ⩽ c . (4.14) Since Q 0 + λ 2 ∥ · ∥ 2 2 is a squared norm on D [ Q 0 ], the first condition in (4.14) ensures that, up to extraction of a subsequence, { φ N ,λ } N ∈ N con v erges weakly in D [ Q 0 ] to φ ∞ ,λ . In other words, by a diagonal argumen t, w e ha v e that the following weak conv ergences in L 2 ( R d ) hold simultaneously: φ N ,λ w − − − − − → N → + ∞ φ ∞ ,λ , ( − i ∇ + A ) φ N ,λ w − − − − − → N → + ∞ ( − i ∇ + A ) φ ∞ ,λ , √ V φ N ,λ w − − − − − → N → + ∞ √ V φ ∞ ,λ . On the other hand, Lemma 4.3 together with the second condition in (4.14) imply that 1 N P N j =1 | p j | 2 ⩽ c γ λ . W e can now apply Lemma 3.1, to deduce that the sequence of complex measures ν N = 1 N N X j =1 p j δ x j , 16 DOMENICO CAFIERO, MICHELE CORREGGI, AND D A VIDE FERMI admits a w eakly con v ergen t subsequence. F urthermore, the limit measure ν ∞ ∈ M C ( R d ) is absolutely con tin uous with resp ect to the density µ ∞ in tro duced in Assumption 1, and d ν ∞ = p d µ ∞ , for some p ∈ L 2 ( R d , d µ ∞ ) . W e now observe that D ( H 0 ) = H 2 ⊂ H 2 loc ( R d )  → C 0 ( R d ) (see Remark 2.2 and Lemma 3.5). F urther- more, considering that the supp orts of the measures ν N and ν ∞ are all contained within a fixed b ounded set, w e hav e that ν N ∈ ( H 2 ) ′ ≡ H − 2 , for b oth d = 2 and d = 3 (see Lemma 3.4). Hence, by Lemma 3.6, R λ 0 ν N ∈ L 2 ( R d ) and, for any f ∈ L 2 ( R d ), D R λ 0 ν N    f E = 1 N N X j =1 p j ( R λ 0 f )( x j ) = 1 N N X j =1 p j ˆ R d G λ 0 ( x j , y ) f ( y ) d y = D 1 N P N j =1 p j G λ 0 ( · , x j )    f E , whic h sho ws that  R λ 0 ν N  ( x ) = 1 N N X j =1 p j G λ 0 ( x , x j ) . (4.15) In addition, from the weak conv ergence of ν N it follows that lim N → + ∞ D R λ 0 ν N    f E = D R λ 0 ν ∞    f E , (4.16) whic h, b y the same arguments outlined ab ov e and Assumption 3, implies that  R λ 0 ν ∞  ( x ) = ˆ R d G λ 0 ( x , y ) p ( y ) U ( y ) d y =  R λ 0 U p  ( x ) . Notice that, since U ∈ L ∞ ( R d ) has compact supp ort and p ∈ L 2 ( R d , U d x ), we hav e indeed p ∈ L 2 ( R d ), U p ∈ L 2 ( R d ) and R λ 0 U p ∈ H 2 = D ( H 0 ) ⊂ D [ Q 0 ]. Summing up, for an y weakly con vergen t sequence { ψ N } N ∈ N satisfying (4.13), we get ψ N = φ N ,λ + 1 N N X j =1 p j G λ 0 ( · , x j ) w − − − − → N → ∞ ψ ∞ = φ ∞ ,λ + R λ 0 U p ∈ D [ Q 0 ] . Recalling the definition of Q λ ∞ in (2.14) and (4.2), by direct calculations, w e further obtain Q λ ∞ [ ψ ∞ ] = ∥ ( − i ∇ + A ) ψ ∞ ∥ 2 2 +    √ V ψ ∞    2 2 + λ 2 ∥ ψ ∞ ∥ 2 2 −  ψ ∞   U a   ψ ∞  = ∥ ( − i ∇ + A ) φ ∞ ,λ ∥ 2 2 + ∥ √ V φ ∞ ,λ ∥ 2 2 + λ 2 ∥ φ ∞ ,λ ∥ 2 2 + 2 ℜ  φ ∞ ,λ , ( H 0 + λ 2 ) R λ 0 U p  + ⟨ R λ 0 U p, ( H 0 + λ 2 ) R λ 0 U p ⟩ −  ψ ∞ , U a ψ ∞  = Q 0 [ φ ∞ ,λ ] + λ 2 ∥ φ ∞ ,λ ∥ 2 2 − D U p    R λ 0 U p E + ⟨ p | a U p ⟩ −    √ a U p − q U a ψ ∞    2 2 . No w, w e can write Q λ N [ ψ N ] − Q λ ∞ [ ψ ∞ ] =  Q 0 [ φ N ,λ ] + λ 2 ∥ φ N ,λ ∥ 2 2  −  Q 0 [ φ ∞ ,λ ] + λ 2 ∥ φ ∞ ,λ ∥ 2 2  + 1 N 2 P i  = j p i  Ξ λ N  ij p j +  U p   R λ 0 U p  + 1 N 2 P N j =1  Ξ λ N  j j | p j | 2 − ⟨ p | a U p ⟩ +    √ a U p − q U a ψ ∞    2 2 . (4.17) Let us discuss separately the addenda app earing on the right-hand side of (4.17). Firstly , from the weak con v ergence of { φ N ,λ } N ∈ N to φ ∞ ,λ in D [ Q 0 ] and the weak low er-semicon tin uit y of the norm, we readily get lim inf N → ∞  Q 0 [ φ N ,λ ] + λ 2 ∥ φ N ,λ ∥ 2 2  ⩾ Q 0 [ φ ∞ ,λ ] + λ 2 ∥ φ ∞ ,λ ∥ 2 2 . HOMOGENIZA TION OF POINT INTERACTIONS 17 Secondly , recalling (3.9) and noting that ϑ A ( x , y ) ∈ C ∞ ( R d × R d ) and h λ ( x , y ) ∈ C 0 ( R d × R d ), by Lemma 4.1 and the weak conv ergence of { ν N } N ∈ N to ν ∞ in M C ( R d ), we infer lim N → + ∞ 1 N 2 P i  = j p i  Ξ λ N  ij p j = − lim N → + ∞ 1 N 2 P i  = j p i h ζ 0 ( x i − x j ) e iϑ A ( x i , x j ) + h λ ( x i , x j ) i p j = − ˆ R d × R d  ζ 0 ( x − y ) e iϑ A ( x , y ) + h λ ( x , y )  d ν ∞ ( x ) d ν ∞ ( y ) = − D U p    R λ 0 U p E . (4.18) Thirdly , notice that the w eak conv ergence of the complex measures ν N , together with the contin uit y and b oundedness of a (see Assumption 3), implies lim N → + ∞ 1 N N X j =1 p j q a ( x j ) ϕ ( x j ) = ˆ R d √ a ϕ d ν ∞ , for any ϕ ∈ C b ( R d ) . By Cauc h y-Sc hw artz inequalit y and the weak conv ergence of { µ N } N ∈ N to µ ∞ (see Assumption 1), it follo ws that     √ a p   ϕ  L 2 ( R d , d µ ∞ )    ⩽  lim N → ∞ 1 N P N j =1 a ( x j ) | p j | 2  1 / 2 ∥ ϕ ∥ L 2 ( R d , d µ ∞ ) . Hence, ⟨ p | a U p ⟩ =   √ a p   2 L 2 ( R d , d µ ∞ ) = sup 0  = φ ∈ C b ( R d )    ⟨ √ a p | ϕ ⟩ L 2 ( R d , d µ ∞ )    2 ∥ ϕ ∥ L 2 ( R d , d µ ∞ ) ⩽ lim N → ∞ 1 N P N j =1 a ( x j ) | p j | 2 = lim N → ∞ 1 N 2 P N j =1 α j | p j | 2 ⩽ lim N → ∞ 1 N 2 P N j =1  Ξ λ N  j j | p j | 2 , (4.19) b y Assumption 3 and the explicit form of the diagonal en tries of the matrix Ξ λ N . No w, dropping the last addendum on the righ t-hand side of (4.17) thanks to its positivity and exploiting that lim inf N →∞ ( s N + t N ) ⩾ lim inf N →∞ s N + lim inf N →∞ t N , we combine (4.17)-(4.19), to obtain that lim inf N → ∞ Q λ N [ ψ N ] − Q λ ∞ [ ψ ∞ ] ⩾ 0 , (4.20) whic h finally prov es the claim (4.12). □ 4.2. Pro of of the Γ − lim sup (in)equalit y. W e now turn to the pro of of item (ii) in Theorem 2.1. I n this connection, we recall that it suffices to prov e the claim for ψ ∞ ∈ D ( H ∞ ), where H ∞ is the selfadjoint op erator asso ciated to Q ∞ (see condition (ii’) in § 3.1 and the related commen ts). First of all, k eeping in mind that D ( H ∞ ) ≡ D ( H 0 ) and recalling the identit y (2.11), w e pro ceed to construct a so-called r e c overy se quenc e { ψ N } N ∈ N for any given ψ ∞ ∈ D ( H 0 ). Lemma 4.4. L et Assumptions 1 and 3 hold. L et ψ ∞ ∈ D ( H 0 ) and c onsider the se quenc e of functions given by ψ N ( x ) := φ λ ( x ) + 1 N N X j =1 p ( x j ) G λ 0 ( x , x j ) , (4.21) wher e p := 1 a ψ ∞ ∈ C 0 ( R d ) , and φ λ := ψ ∞ − R λ 0 U p ∈ D ( H 0 ) . (4.22) Then, the fol lowing pr op erties hold: (i) ψ N ∈ D [ Q λ N ] , for al l N ∈ N ; (ii) { ψ N } N ∈ N c onver ges in norm to ψ ∞ , as N → ∞ . 18 DOMENICO CAFIERO, MICHELE CORREGGI, AND D A VIDE FERMI Pr o of. (i) Since ψ ∞ ∈ D ( H 0 ) = H 2 ⊂ C 0 ( R d ) by Lemma 3.5, Assumption 3 implies that p = ψ ∞ /a is a contin uous function on R d . So, the co efficien ts p ( x j ) ∈ C app earing in (4.21) are well-defined. T o say more, considering that U ∈ L ∞ ( R d ) has compact supp ort by Assumption 1, w e ha v e that U p ∈ L 2 ( R d ) and, as a consequence, φ λ = ψ ∞ − R λ 0 U p ∈ D ( H 0 ) ⊂ D [ Q 0 ]. In view of (2.2) and (4.1), the ab ov e argumen ts suffice to infer that ψ N ∈ D [ Q λ N ] for any N ∈ N . (ii) By (4.21) and (4.22), ψ N ( x ) − ψ ∞ ( x ) = 1 N P N j =1 p ( x j ) G λ 0 ( x , x j ) −  R λ 0 U p  ( x ) . W e can then repro duce the argument in the pro of of Prop osition 4.1 (see (4.16) and the related commen ts) to show that the right-hand side of the ab o v e expression conv erges w eakly to 0 in L 2 ( R d ). Then, to conclude the pro of we just hav e to pro ve that lim N →∞    1 N P N j =1 p ( x j ) G λ 0 ( · , x j )    2 =    R λ 0 U p    2 , (4.23) but    1 N P N j =1 p ( x j ) G λ 0 ( · , x j )    2 2 = 1 N 2 P N i,j =1 p ( x i ) p ( x j ) ˆ R d G λ 0 ( x i , y ) G λ 0 ( y , x j ) d y and the integral in the ab ov e expression actually coincides with the integral kernel of R λ 0 2 ev aluated at ( x i , x j ). Considering that this k ernel is indeed a b ounded and contin uous function on R d × R d [BGP07, Thm. 21, item (5)], and that p is con tinuous on R d , b y w eak conv ergence of pro duct measures [Bo18, Prop. 2.7.8], w e readily obtain lim N →∞    1 N P N j =1 p ( x j ) G λ 0 ( · , x j )    2 2 = ˆ R d × R d p ( x ) p ( y )  ˆ R d G λ 0 ( x , y ′ ) G λ 0 ( y ′ , y ) d y ′  U ( x ) U ( y ) d x d y = D U p    R λ 0 2 U p E =    R λ 0 U p    2 2 , whic h pro v es (4.23), whence the thesis. □ W e are now in a p osition to establish the limit sup erior identit y . Prop osition 4.2. L et Assumptions 1 to 3 hold and let λ > 0 . Then, for any ψ ∞ ∈ D ( H 0 ) ther e exists a se quenc e { ψ N } N ∈ N such that ψ N ∈ D [ Q λ N ] for al l N ∈ N , ψ N − − − − − → N → + ∞ ψ ∞ , and lim N → ∞ Q λ N [ ψ N ] = Q λ ∞ [ ψ ∞ ] . (4.24) Pr o of. Let us consider the reco very sequence { ψ N } N ∈ N in tro duced in Lemma 4.4. W e only ha v e to establish the identit y (4.24). Noting that ψ ∞ , R λ 0 U p ∈ D ( H 0 ), in tegrating b y parts and using the basic identit y ( H 0 + λ 2 ) R λ 0 U p = U p ∈ L 2 ( R d ), we obtain Q λ N [ ψ N ] − Q λ ∞ [ ∞ ] =   ( − i ∇ + A )( ψ ∞ − R λ 0 U p )   2 2 +   √ V ( ψ ∞ − R λ 0 U p )   2 2 + λ 2   ψ ∞ − R λ 0 U p   2 2 + 1 N 2 P N i,j =1 p ( x i )  Ξ λ N  ij p ( x j ) − h ∥ ( − i ∇ + A ) ψ ∞ ∥ 2 2 +   √ V ψ ∞   2 2 + λ 2 ∥ ψ ∞ ∥ 2 2 −  ψ ∞   U a   ψ ∞  i = − ⟨ p | a U p ⟩ + D U p    R λ 0 U p E + 1 N 2 P N j =1  Ξ λ N  j j | p ( x j ) | 2 + 1 N 2 P i  = j p ( x i )  Ξ λ N  ij p ( x j ) . Since b oth a and p are contin uous by Assumption 3 and (4.22), the weak con vergence of measures in Assumption 1 yields lim N → ∞ 1 N 2 P N j =1  Ξ λ N  j j | p ( x j ) | 2 = lim N → ∞ 1 N 2 P N j =1  N a ( x j ) + O (1)  | p ( x j ) | 2 = ˆ R d a | p | 2 d µ ∞ = ⟨ p | a U p ⟩ . HOMOGENIZA TION OF POINT INTERACTIONS 19 On the other hand, b y arguments analogous to those rep orted in the pro of of Prop osition 4.1 (see, in particular, (4.18) and the related comments), w e infer that lim N →∞ 1 N 2 N X i,j =1 , i  = j p ( x i )  Ξ λ N  ij p ( x j ) = − D U p    R λ 0 U p E , whic h completes the pro of. □ 4.3. Pro of of the uniform resolven t con v ergence. In this conclusiv e section w e presen t the pro of of Corollary 2.1. Also in this case, we firstly state an auxiliary result on the compactness prop erties of the extended resolven t op erator R λ 0 in tro duced in Lemma 3.6. Lemma 4.5. L et λ > 0 . If the r esolvent R λ 0 is c omp act on L 2 ( R d ) , then it extends to a c omp act op er ator R λ 0 : H r → H r +2 − ε , for al l r ∈ R , ε > 0 . (4.25) Pr o of. By the compactness of R λ 0 as an op erator acting in L 2 ( R d ), we infer that H 0 has purely discrete sp ectrum σ ( H 0 ) =  λ 2 ℓ  ℓ ∈ N , with λ ℓ ⩾ λ ℓ +1 and λ ℓ → ∞ as  → ∞ [Sc12, Prop. 5.12]. In particular, we ha v e the iden tit y R λ 0 ψ = P ℓ ∈ N 1 λ 2 ℓ + λ 2 ⟨ b ℓ | ψ ⟩ b ℓ , where { b ℓ } ℓ ∈ N is an orthonormal basis of eigenv ectors of H 0 . It is easy to chec k that b ℓ ∈ H r for an y r ∈ R . Let us now introduce the sequence of truncated op erators R λ L ψ = P ℓ ⩽ L 1 λ 2 ℓ + λ 2 ⟨ b ℓ | ψ ⟩ b ℓ , for L ∈ N . These op erators ha v e finite-rank, so they are compact. T o infer the thesis, it suffices to show that R λ L − R λ 0 → 0 as L → ∞ , with resp ect to the top ology of b ounded op erators from H r to H r +2 − ε , for ε > 0. In this connection, let us p oint out that    ( R λ 0 − R λ L ) ψ    H r +2 − ε =    P ℓ ⩾ L ( λ 2 ℓ +1) r/ 2+1 − ε/ 2 λ 2 ℓ + λ 2 ⟨ b ℓ | ψ ⟩ b ℓ    2 ⩽ 1 ( λ 2 L + λ 2 ) ε   P ℓ ⩾ L ( λ 2 ℓ + 1 ) r/ 2 ⟨ b ℓ | ψ ⟩ b ℓ   2 ⩽ 1 ( λ 2 L + λ 2 ) ε ∥ ψ ∥ H r . Recalling that λ L → ∞ for L → ∞ , this implies that    R λ 0 − R λ L    B ( H r , H r +2 − ε ) ⩽ 1 ( λ 2 L +1) ε − − − − − → L → ∞ 0 , ultimately proving the thesis. □ W e now pro ve the claim stated in Corollary 2.1. Pr o of of Cor ol lary 2.1. The first claim follows by Theorem 2.1 and standard argumen ts of Γ − con v ergence theory [DM93, § 13]. The lifting of the conv ergence to norm resolven t sense, whenever the resolven t R λ 0 = ( H 0 + λ 2 ) − 1 is compact for some λ > 0 is t ypically taken from gran ted, but it is in fact a delicate question in general. Indeed, v ery often, one can exploit a uniform con trol on the sequence of resolv en ts ( H N + λ 2 ) − 1 , lik e e.g. when the domains of the op erators or of the quadratic forms are indep endent of N , and appro ximate them b y finite rank op erators, which allows to deduce con v ergence in norm. This, ho w ever, is certainly not obvious when, like in our case, the form domains dep end on N . Therefore, we pro vide here a detailed pro of of the statement. The second claim indeed follo ws from Theorem 3.2, once we assess asymptotic compactness of the forms Q N . Namely , we hav e to sho w that for any sequence { ψ N } N ∈ N ⊂ L 2 ( R d ) satisfying lim inf N →∞  Q N [ ψ N ] + λ 2 ∥ ψ N ∥ 2 2  < + ∞ , (4.26) 20 DOMENICO CAFIERO, MICHELE CORREGGI, AND D A VIDE FERMI for λ > 0 sufficien tly large, one can extract a subsequence which con verges strongly in L 2 ( R d ) (see (3.2)), that will still b e lab el by N ∈ N . Arguing as in the pro of of Prop osition 4.1, we may infer that the condition (4.26) ensures that there exists a finite constant c > 0 suc h that Q 0 [ φ N ,λ ] + λ 2 ∥ φ N ,λ ∥ 2 2 ⩽ c , 1 N 2 P N j =1 | p j | 2 ⩽ c . On the one hand, the compactness of R λ 0 implies that the op erator ( H 0 + λ 2 ) − 1 / 2 is compact, as well. Considering that ∥ ( H 0 + λ 2 ) 1 / 2 ψ N ∥ 2 2 yields an equiv alent norm on D [ Q 0 ], by classical argumen ts, we get that D [ Q 0 ] is compactly em b edded in L 2 ( R d ) (see, e.g. , [Sc12, Prop. 5.12]). Since φ N ,λ is uniformly b ounded in D [ Q 0 ], we may extract a strongly conv ergen t subsequence. On the other hand, let us refer once more to the sequence of measures ν N := 1 N P N j =1 p j δ x j , for N ∈ N , and recall that, by (4.15), 1 N P N j =1 q j G λ 0 ( · , x j ) = R λ 0 ν N . Thanks Lemma 3.5, for any ε ∈ (0 , 2 − d/ 2) and f ∈ H 2 − ε ⊂ C 0 ( R d ), we hav e |⟨ ν N | f ⟩| ⩽ 1 N P N j =1 | p j | | f ( x j ) | ⩽  1 N P N j =1 | p j | 2  1 / 2 sup x ∈ supp U | f ( x ) | ⩽ c 1 / 2 ∥ f ∥ H 2 − ε . Hence, the sequence ν N is uniformly b ounded in ( H 2 − ε ) ′ = H − (2 − ε ) (see Lemma 3.4) and b y Lemma 4.5, w e infer that 1 N P N j =1 q j G λ 0 ( · , x j ) conv erges strongly in L 2 ( R d ), up to the extraction of a subsequence. Putting together the ab ov e results, by a diagonal argumen t, we obtain that ψ N admits a subsequence whic h con v erges strongly in L 2 ( R d ), thus concluding the pro of. □ References [A GH+88] S. Alb everio, F. Gesztesy , R. Høegh-Krohn, H. Holden, Solv able Mo dels in Quantum Mechanics , Springer, Berlin- Heidelb erg (1988). [AK00] S. Alb everio, P . Kurasov, Singular p erturbations of differential operators , London Mathematical So ciety Lecture Note Series 271 , Cambridge Universit y Press, Cam bridge (2000). [AKN07] S. Alb everio, S. Kuzhel’, L. Nizhnik, Singularly p erturb ed self-adjoin t op erators in scales of Hilb ert spaces , Ukr. Math. J. 59 , 787–810 (2007). [BD93] L. B´ aez-Duarte, Cen tral limit theorem for complex measures , J. Theor. Prob. 6 , 33–56 (1993). [BCT18] G. Basti, S. Cenatiemp o, A. T eta, Univ ersal low-energy b ehavior in a quan tum Lorentz gas with Gross-Pitaevskii p oten tials , Math. Phys. Anal. Geom. 21 , Art. 11 (2018). [BdO+14] R. Bedo ya, C. R. de Oliveira, A. A. V erri, Complex Γ -conv ergence and magnetic Diric hlet Laplacian in b ounded thin tub es , J. Sp ectr. Theory 4 (3), 621–642 (2014). [BF61] F.A. Berezin, L.D. F addeev, A remark on Schr¨ odinger’s equation with a singular p otential , Dokl. Ak ad. Nauk Ser. Fiz. 137 (5), 1011–1014 (1961); translation in Sov. Math. Dokl. 2 , 372–375 (1961). [BP35] H. Bethe, R. Peierls, Quan tum theory of the diplon , Pro c. R. Soc. Lond. Ser. A Math. Ph ys. Eng. Sci. 148 (863), 146–156 (1935). [Bo18] V.I. Bogachev, W eak con vergence of measures , Mathematical Surveys and Monographs 234 , AMS, Providence, RI (2018). [BK13] J. Bolte, S. Kepp eler, Heat kernel asymptotics for magnetic Schr¨ odinger op erators , J. Math. Phys. 54 , 112104 (2013). [Br02] A. Braides, Γ -conv ergence for beginners , Oxford Universit y Press, Oxford (2002). [BFT98] J.F. Brasche, R. Figari, A. T eta, Singular Sc hr¨ odinger Op erator as Limits of P oint Interaction Hamiltonians , P otential Analysis 8 , 163–178 (1998). [BGP05] J. Br ¨ uning, V. Geyler, K. P ankrashkin, On-diagonal singularities of the Green functions for Sc hr¨ odinger op erators , J. Math. Ph ys. 46 , 113508 (2005). [BGP07] J. Br ¨ uning, V. Geyler, K. Pankrashkin, Contin uit y prop erties of integral kernels asso ciated with Schr¨ odinger op erators on manifolds , Ann. H. Poincar ´ e 8 , 781–816 (2007). [CDF08] M. Correggi, G. Dell’Antonio, D. Finco, Sp ectral analysis of a tw o b o dy problem with zero-range p erturbation , J. F unct. Anal. 255 (2), 502–531 (2008). HOMOGENIZA TION OF POINT INTERACTIONS 21 [CP82] J. Chazarain, A. Piriou, Introduction to the theory of linear partial differential equations , Studies in Mathematics and its Applications 14 , North-Holland Publishing Co., Amsterdam-New Y ork (1982). [DFT94] G.F. Dell’Antonio, R. Figari, A. T eta, Hamiltonians for systems of N particles interacting through point in terac- tions , Ann. Inst. Henri Poincar ´ e 60 (3), 253–290 (1994). [DFT05] G. Dell’Antonio, D. Finco, A. T eta, Singularly p erturb ed Hamiltonians of a quan tum Rayleigh gas defined as quadratic forms , P otential Anal. 22 (3), 229–261 (2005). [DM93] G. Dal Maso, An introduction to Γ -con vergence , Progress in Nonlinear Differential Equations and their Applica- tions 8 , Birkh¨ auser, Boston (1993). [DeO09] C.R. de Oliveira, Intermediate sp ectral theory and quantum dynamics , Progress in Mathematical Physics 54 , Birkh¨ auser, Basel (2009). [DeO11] C.R. de Oliveira, Quan tum singular op erator limits of thin Diric hlet tub es via Γ -con vergence , Rep. Math. Phys. 67 (1), 1–32 (2011). [EN03] P . Exner, K. Nˇ emcov´ a, Leaky quantum graphs: approximations by point-in teraction Hamiltonians , J. Phys. A: Math. Gen. 36 (40), 10173–10193 (2003). [F e16] D. F ermi, A functional analytic framework for lo cal zeta regularization and the scalar Casimir effect , PhD thesis, Do ctoral Sc ho ol in Mathematics, 28 th cycle, Univ ersity of Milan (2016). [FP17] D. F ermi, L. Pizzo cchero, Lo cal zeta regularization and the scalar Casimir effect. A general approach based on in tegral kernels , W orld Scientific, Singapore (2017). [FP23] D. F ermi, L. Pizzo cchero, On the Casimir effect with δ -like p otentials, and a recent pap er by K. Ziemian (Ann. H. P oincar´ e, 2021) , Ann. H. Poincar ´ e 24 , 2363–2400 (2023). [F e36] E. F ermi, Sul moto dei neutroni nelle sostanze idrogenate , Ric. Scientifica 7 (2), 13–52 (1936). [FHT98] R. Figari, H. H¨ olden, A. T eta, A law of large num bers and a central limit theorem for the Schr¨ odinger operator with zero-range p oten tials , J. Stat. Phys. 51 (1/2), 205–214 (1998). [F OT85] R. Figari, E. Orlandi, A. T eta, The Laplacian in regions with many small obstacles: fluctuations around the limit op erator , J. Stat. Ph ys. 41 (3-4), 465–487 (1985). [F OT87] R. Figari, E. Orlandi, A. T eta, A central limit theorem for the Laplacian in regions with many small holes , in “Sto c hastic Pro cesses - Mathematics and Physics I I (Bielefeld, 1985)”, 75–86, Lecture Notes in Math. 1250 , Springer, Berlin (1987). [FT92] R. Figari, A. T eta, Effective p otential and fluctuations for a b oundary v alue problem on a randomly perforated domains , Lett. Math. Phys. 26 , 295–305 (1992). [FT93] R. Figari, A. T eta, A b oundary v alue problem of mixed type on p erforated domains , Asympt. Anal. 6 (3), 271–284 (1993). [F o99] G.B. F olland, Real Analysis, Mo dern T echniques and Their Applications (2 nd ed.) , John Wiley and Sons, New Y ork (1999). [KS03] K. Kuw ae, T. Shioy a, Conv ergence of sp ectral structures: a functional analytic theory and its applications to sp ectral geometry , Comm. Anal. Geom. 11 (4), 599–673 (2003). [LL01] E.H. Lieb, M. Loss, Analysis (2 nd ed.), AMS Graduate Studies in Mathematics 14 , AMS, Providence (2001). [Mo94] U. Mosco, Comp osite Media and Asymptotic Dirichlet F orms , J. F unct. Anal. 123 (2), 368–421 (1994). [OLB+10] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, NIST Handb o ok of mathematical functions , Cambridge Univ ersity Press, Cambridge (2010). [Oz06] K. Ozano v a, Approximation by point p otentials in a magnetic field , J. Phys. A: Math. Gen. 39 (12), 3071–3083 (2006). [Oz83] S. Ozaw a, Poin t in teraction potential appro ximation for ( − ∆ + U ) − 1 and eigen v alues of the Laplacian on widely p erturb ed domain , Osak a J. Math. 20 , 923–937 (1983). [PV80] G. Papanicolau, S. V aradhan, Diffusion in regions with many small holes , pp. 190-206 in B. Grigelionis (eds) “Sto c hastic Differential Systems Filtering and Control”, Lecture Notes in Control and Information Sciences 25 , Springer, Berlin, Heidelb erg (1980). [P o01] A. Posilicano, A Krein-lik e formula for singular p erturbations of self-adjoint op erators and applications , J. F unct. Anal. 183 (1), 109–147 (2001). [R T75] J. Rauch, M. T aylor, Poten tial and scattering theory on wildly p erturb ed domains , J. F unct. Anal. 18 , 27–59 (1975). [Ru87] W. Rudin, Real and Complex Analysis (3 rd ed.), McGra w-Hill Bo ok Co., Singap ore (1987). [Sc12] K. Schm¨ udgen, Unbounded Self-adjoint Op erators on Hilb ert Space , Springer Graduate T exts in Mathematics 265 , Springer Science, Dordrech t (2012). [Se15] S. Serfaty , Coulomb Gases and Ginzburg-Landau V ortices , Zurich Lectures in Adv anced Mathematics, EMS, EMS Press (2015). [T e90] A. T eta, Quadratic forms for singular p erturbation of the Laplacian , Publ. R.I.M.S. Ky oto Univ. 26 , 803–817 (1990). 22 DOMENICO CAFIERO, MICHELE CORREGGI, AND D A VIDE FERMI [Th35] L.H. Thomas, The Interaction Betw een a Neutron and a Proton and the Structure of H3 , Phys. Rev. 47 (12), 903–909 (1935). Dip ar timento di Ma tema tica, Politecnico di Milano, P.zza Leonardo da Vinci, 32, 20133, Milano, It al y Email address : domenico.cafiero@polimi.it Dip ar timento di Ma tema tica, Politecnico di Milano, P.zza Leonardo da Vinci, 32, 20133, Milano, It al y Email address : michele.correggi@polimi.it URL : https://sites.google.com/view/michele-correggi Dip ar timento di Ma tema tica, Politecnico di Milano, P.zza Leonardo da Vinci, 32, 20133, Milano, It al y, and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, It al y Email address : davide.fermi@polimi.it URL : https://fermidavide.com