From Short-Range to Contact Interactions in Two-dimensional Many-Body Quantum Systems

Quantum systems composed of $N$ distinct particles in $\R^2$ with two-body contact interactions of TMS type are shown to arise as limits - in the norm resolvent sense - of Schr\"odinger operators with suitably rescaled pair potentials.


Introduction
Many-particle quantum systems with short-range interactions are often described by simplified models with zero-range (contact) interactions. Effective models of low energy nuclear physics, the Lieb-Liniger model for the 1d Bose gas, and the Fermi polaron model for an impurity within an ideal Fermi gas are well-known models of this type [25,14,18]. The understanding of these models is that zero-range interactions provide an idealized description of short-ranged interparticle forces whose details are not known or considered irrelevant. In the present paper we fully justify this idealization for 2d many-particle systems with (spin-independent) two-body forces. We prove norm resolvent convergence for suitably rescaled Schrödinger operators towards TMS Hamiltonians [6]. This mathematically justifies the use of TMS Hamiltonians for describing contact interactions, and, moreover, provides a new way of defining them.
For H ε to have a limit as ε → 0, the coupling constant g ε,σ must have an asymptotic behavior, as ε → 0, adjusted to the space dimension. For d = 2 we may set with a σ , b σ ∈ R, a σ > 0. Here the reduced mass µ σ of the pair σ = (i, j) has been factored out. We assume that a σ ≥ 1 2π V σ (r) dr. (1.4) If equality holds in (1.4), we will see that the two-body interaction V σ gives rise to a non-trivial contact interaction, whose strength is determined by the parameter b σ . In the case of inequality there is no contribution from V σ . This is well-known for the case of N = 1 particles, where our problem was solved long ago [2]. We use J ⊂ I to denote the subset of pairs for which equality holds in (1.4).
Our main result asserts norm resolvent convergence of H ε as ε → 0. In addition, we give an explicit expression for the resolvent of the limiting Hamiltonian, a characterization of the domain in terms of so-called TMS boundary conditions at the collision planes {x ∈ R 2N | x i = x j }, and we show, in the case of equal masses m j = 1, that our limiting Hamiltonian agrees with the Hamiltonian H β (for suitable β = (β σ ) σ∈I ) of Dell'Antonio et al. (see [6,Eqs.  Then G(z) σ = T σ R 0 (z), with R 0 (z) = (H 0 + z) −1 , is a bounded operator from H to X σ . This operator is closely related to the Green's function of H 0 +z, see, e.g., (4.52), below, and the letter G serves to remind us of the connection. Let T and G(z) = T R 0 (z) denote the operator-valued vectors with components T σ and G(z) σ , respectively. Our main result is the following: * Theorem 1.1. Suppose, for all σ ∈ I, that V σ ∈ L 1 ∩ L 2 (R 2 ), V σ (−r) = V σ (r), and there exists some s > 0 such that |r| 2s |V σ (r)| dr < ∞. Let H ε be defined by (1.1)- (1.4). Then, as ε → 0, H ε converges in norm resolvent sense to a self-adjoint, semibounded operator H. The resolvent of H, for z ∈ ρ(H 0 ) ∩ ρ(H), obeys with some invertible and unbounded operator Θ(z) : D ⊂ X → X, whose domain is independent of z.
2. Antisymmetric wave functions from H 2 (R 2N ) belong to the kernel of all trace operators T σ . Hence for identical particles, by (1.7), (H + z) −1 = R 0 (z) on the subspace H anti ⊂ H of antisymmetric wave functions. In fact, there are no contact interactions among identical, spin-aligned fermions in space dimensions d ≥ 2 [10]. Theorem 1.1 implies lim ε→0 (H ε + z) −1 = R 0 (z) on H anti and its proof gives a rate of convergence [10].
The main novelty in Theorem 1.1, compared to previous results of similar type, is that convergence is established in norm resolvent sense. Norm resolvent convergence implies convergence of the spectrum, which is not true for the weaker strong resolvent convergence [24]. In the present case, where σ(H ε ) = [Σ ε , ∞), the norm resolvent convergence implies σ(H) = [Σ, ∞) with Σ = lim ε→0 Σ ε .
The analog of Theorem 1.1 in the case N = 1 is well-known and true without a condition of the type (1.4). But if N > 1 this condition is necessary: if (1.4) is not satisfied for some pair σ, then one may expect strong resolvent convergence at best. To see this, consider a twoparticle system with pair potential V and reduced mass µ = 1. If 0 < a < 1 2π V (r) dr and −g −1 ε = a ln(ε) + b, then the Hamiltonian in the center of mass frame has a negative eigenvalue running towards −∞ as ε → 0 [23]. Due to the center of mass motion, the Hamiltonian H ε then has essential spectrum filling the entire real axis in the limit ε → 0. This is not compatible with norm resolvent convergence towards a semibounded Hamiltonian.
By the following corollary, domain vectors of H satisfy the so-called TMS boundary conditions at the collision planes {x ∈ R 2N | x i = x j }. The vectors ψ 0 and w are uniquely determined by ψ ∈ D(H) and z ∈ ρ(H 0 ) ∩ ρ(H).
Choosing w = 0 in Corollary 1.2, we see that D(H 0 ) ∩ Ker T ⊂ D(H) and that H = H 0 on D(H 0 ) ∩ Ker T . Thus H is a self-adjoint extension of H 0 ↾ Ker T . It is well-known that a Krein formula, like (1.7), is the characteristic equation for the resolvent of all such extensions [19,20].
Recently, in a study of the 2d stochastic heat equation, Gu, Quastel and Tsai have derived a result very similar to Theorem 1.1 for N identical particles [12]. In [12] the two-body potentials are compactly supported smooth functions and convergence in strong resolvent sense is established. In the case of bosons, Gu et al. give a formula for the resolvent that is similar to the one in Theorem 1.1 and the subsequent remark. In one space dimension, results similar to Theorem 1.1 for N = 3 quantum particles are obtained in [3], and for arbitrary N ≥ 1 in [11]. The well-known Lieb-Liniger model with repulsive δ-interactions is derived from a trapped 3d Bose gas with two-body potentials in [22]. TMS Hamiltonians like H in Theorem 1.1 have also been described as resolvent limits of N -body Hamiltonians, where the regularized two-body contact interaction is an integral operator, rather than a potential, and the regularization is achieved by an ultraviolet cutoff [6,7,9] or a reversed heat flow [8]. In these cases the convergence is easier to establish than in the case studied here. Nevertheless, all previous approximation results of this kind in 2d with N ≥ 2 particles establish strong resolvent convergence only. In three space dimensions, TMS Hamiltonians for N ≥ 3 bosons are symmetric but not self-adjoint, and all (non-trivial) self-adjoint extensions are unbounded below [17,16]. A possible solution to this problem in the case of N = 3 bosons is worked out in the recent paper [4], where a semi-bounded self-adjoint Hamiltonian containing a three-body contact interaction is constructed.
To prove Theorem 1.1, we further develop the methods and tools introduced in our previous paper [11]. The key elements along with some auxiliary spaces and tools are described in Section 2. Sections 3 and 4 provide all preparations needed for the proof of Theorem 1.1, which is given in Section 5. In addition, we derive a lower bound on σ(H) in Section 5. In Section 6 we compute the quadratic form of the Hamiltonian H and we show it agrees, in the case of equal masses, with a quadratic form derived by Dell'Antonio et al. [6], and which, in recent years, has become a standard starting point for investigations of TMS Hamiltonians.
We conclude this introduction with some remarks on our notations and with the proof of Corollary 1.2. In this paper the resolvent set ρ(H) of a closed operator H is defined as the set of z ∈ C for which H + z : D(H) ⊂ H → H is a bijection. This differs by a minus sign from the conventional definition and it means that the spectrum σ(H) is the complement of −ρ(H). The L 2 -norm will be denoted by · , without index, while all other norms carry the space as an index, as e.g. in V L 1 .
Proof of Corollary 1.2. In this proof, T : D(H 0 ) → X and G(z) : H → X are the operatorvalued vectors defined in terms of the components T σ and G(z) σ . We know from [5] and Proposition 5.1 that a point z ∈ ρ(H 0 ) belongs to ρ(H) if and only if Θ(z) has a bounded inverse.

Auxiliary operators and strategy of the proof
The proof of Theorem (1.1) starts with the new expression (2.9), below, for H ε , which allows for the explicit representation of the resolvent in terms of a generalized Konno-Kuroda formula, see Equation (2.11). We now describe the various operators occurring in (2.9) and (2.11). To this end, we need the auxiliary Hilbert spaces (2.1) The integration variables r and R in (2.1) correspond to the relative and center of mass coordinates of the particles constituting the pair σ = (i, j). This change of coordinates is implemented unitarily by the operator K (i,j) : H → X (i,j) defined by The adjoint thereof is the operator K * (i,j) : Let U ε ∈ L (L 2 (R 2 )) denote the unitary rescaling that, on L ( X (i,j) ), is given by Then, in terms of the above operators, we define new operators A ε,σ , B ε,σ : where D(A ε,σ ) is determined by the domain of the multiplication operator v σ ⊗ 1. Obviously, A ε,σ and B ε,σ are densely defined and closed. These operators allow us to write the two-body interaction in the form and therefore , and moreover, the hypotheses of Corollary B.2 are satisfied. This means that are bounded operators and that Λ ε (z) has a bounded inverse if and only if z ∈ ρ(H ε ). Then In Section 3 we prove existence of the limit for some, and hence for all z ∈ ρ(H 0 ). This works in all dimensions d ∈ {1, 2, 3}. Obviously, lim ε→0 B ε,σ R 0 (z) = J σ S(z) σ . We will see that where G(z) σ = T σ R 0 (z) and T σ is the trace operator introduced in (1.6). The hard part, which is non-trivial even for N = 1, is the convergence of Λ ε (z) −1 in space dimensions d ≥ 2. The limit is, in general, not the inverse of an operator. Our analysis is based on the decomposition Λ ε (z) = Λ ε (z) diag + Λ ε (z) off into diagonal and off-diagonal parts of the operator matrix (2.10). We then show that both contain a divergent contribution that must be cancelled by the divergence of g −1 ε,σ as ε → 0. It turns out that (g −1 ε,σ − B ε,σ R 0 (z)A * ε,σ ) −1 has a vanishing limit unless V σ (r) dr = 2πa σ . In the end, we arrive at for large enough z > 0, with some closed and invertible operator Θ(z) in the Hilbert space X. Combining (2.14) with (2.12) and (2.13), it follows that the expression on the right hand side of (2.11) has the limit (1.7). It is then a standard argument to show that (1.7) defines the resolvent of a self-adjoint operator.
Proof. Due to the Sobolev embedding H 2 (R 2 ) ֒→ C 0,s (R 2 ), valid for s ∈ (0, 1), we have that This implies the statement of the lemma.
The goal of this section is to show that the limit lim ε→0 Λ ε (z) −1 exists for all large enough z > 0, provided that, for all pairs σ, V σ ∈ L 1 ∩ L 2 (R 2 ) and |r| 2s |V σ (r)| dr < ∞ for some s > 0. As explained at the end of Section 2, this proof is based on a decomposition of Λ ε (z) into diagonal and off-diagonal parts that we now define. Recall from (2.10) that, for given ε, z > 0, the operator matrix Λ ε (z) ∈ L ( X) has the components defines a bounded operator φ ε (z) σν ∈ L ( X ν , X σ ) for all pairs σ, ν ∈ I. Using the definitions (2.7) and (2.6) for B ε,σ and A ε,ν , respectively, leads to the explicit formula We shall see that, for fixed z > 0, the off-diagonal operators φ ε (z) σν (σ = ν) are uniformly bounded in ε > 0, whereas the specific behavior of the coupling constant g ε,σ (see (1.3)) cancels the singular part in the diagonal operators φ ε (z) σσ if and only if a σ = V σ (r) dr/(2π) > 0. Therefore, we write the operator matrix Λ ε (z) in the form where the diagonal part Λ ε (z) diag and the off-diagonal part Λ ε (z) off are given by and respectively. Section 4.1 is devoted to Λ ε (z) diag and Section 4.2 is devoted to Λ ε (z) off .

The limit of
We first show that Λ ε (z) diag is invertible for small enough ε > 0 and large enough z > 0 and then we compute the limit lim ε→0 (Λ ε (z) diag ) −1 . This can be done for each component Λ ε (z) σσ = g −1 ε,σ − φ ε (z) σσ separately and without loss of generality we may choose σ = (1, 2). In the following, v := v (1,2) , We begin by computing the kernel of φ ε (z). To this end, we note that where H 0 is the free Hamiltonian H 0 expressed in the relative and center of mass coordinates of the pair (1, 2) (cf. (2.2)), i.e.
Starting from the explicit formula (4.1), the identity (4.5) and K K * = 1 imply that Hence, after a Fourier transform in (R, x 3 , ..., x N ), φ ε (z) acts pointwise in the conjugate variable P = (P, p 3 , ..., p N ) by the operator where (4.8) and the scaling property of the Laplace operator with respect to the unitary scaling U ε has been used in (4.7). As the resolvent in (4.7) is divergent in the limit ε ↓ 0, the first step in the analysis of lim ε→0 (Λ ε (z) diag ) −1 is to study the asymptotic behavior of φ ε (z, P ) for small ε > 0. A similar but less specific version of the following lemma can be found in [23,Proposition 3.2].
where · HS denotes the Hilbert-Schmidt norm in L 2 (R 2 ), γ is the Euler-Mascheroni constant and L is a Hilbert-Schmidt operator in L 2 (R 2 ) that is defined in terms of the kernel Proof. First, we observe that u(−∆ + α) −1 v is associated with the integral kernel where G α = G 2 α denotes the Green's function of −∆ + α : H 2 (R 2 ) → L 2 (R 2 ) (we refer to the appendix for details about G α ). By [2, Chapter I.5], we have the explicit description where H (1) 0 denotes the Hankel function of first kind and order zero. It is known (see e.g. [1, As G 1 is smooth in R 2 \ {0} and exponentially decaying as |x| → ∞ (see Lemma A.2), we conclude from (4.12) and (4.13) that there is some constant λ = λ(s) > 0 such that Using again (4.12), this implies that . Therefore, (4.9) follows from where the elementary inequality (a + b) 2s ≤ 2 2s (a 2s + b 2s ) (a, b > 0) was used for the last inequality. To show that L indeed defines a Hilbert-Schmidt operator, we note that L 2 HS = I 1 + I 2 , where by the Cauchy-Schwarz inequality and It remains to prove (4.10). To this end, we note that for fixed x = 0 the function α → G α (x) is strictly monotonically decreasing in α > 0 (cf. Lemma A.1 (v)). Therefore, the Cauchy-Schwarz inequality yields for α ≥ 1 the uniform estimate In the next lemma we show that for some a > 0, will play a crucial role.
If V (r) dr/(2π) = a, then we again use (4.14) and (4.19) to derive a more refined version of (4.20): With the help of this bound, the desired estimate for the operator norm An immediate consequence of Lemma 4.2 is that g −1 ε − φ ε (z) is invertible for all small enough ε > 0 and large enough z > 0. Yet, we shall see that the limit ε → 0 of the inverse operator essentially depends on the leading order of the coupling constant g ε or, more precisely, on a. If V (r) dr/(2π) < a, then it follows from (4.15) that lim ε→0 g −1 ε − φ ε (z) −1 vanishes for z > z 0 > 0, while for V (r) dr/(2π) = a this limit will turn out to be non-trivial. To prove this, we first recall that (4.7) and (4.11) imply that for fixed P the operator φ ε (z, P ) is associated with the kernel In the notation of [2, Chapter I.5] for the one-particle case, (−g ε ) · φ ε (z, P ) agrees with the operator B ε (k) for k 2 := −µ(z + Q) < 0, λ 1 := 1/a and λ 2 := −b/a 2 . In view of [2, Chapter I.5, Eq. (5.61)] * , we thus expect that where D(z) is an (unbounded) operator in X σ , which, after passing to Fourier space, acts as the multiplication operator (4.23) Here, γ denotes the Euler-Mascheroni constant and L is the Hilbert-Schmidt operator defined in Lemma 4.1. We now turn to the proof of (4.21): . Let ε 0 , z 0 > 0 be chosen as in Lemma 4.2 and, for the rest of this proof, let z > max(z 0 , z 1 ) be fixed. Then (4.21) is equivalent to the statement that The term in curly braces is to be inverted.
with · denoting the operator norm in L (L 2 (R 2 )). The idea of the following proof is that both operators in the difference vanish as |P | → ∞, while for |P | ≤ K the arguments from the one-particle case still work. To make this more explicit, we fix δ > 0. Then it follows from Lemma 4.2 and (4.22) that provided that ε > 0 is sufficiently small and |P |, and hence Q, are sufficiently large. To prove (4.24), it therefore suffices to show that for any K > 0 there exists ε K > 0 such that holds for all ε ∈ (0, ε K ). Let K > 0 be fixed. Then (4.9) implies that, for some constant Combining this with the asymptotic behavior of g ε in the limit ε → 0 (see (4.14)), we see that is valid in Hilbert-Schmidt norm uniformly in P with Q = Q(P ) ≤ K. As this coincides with the expansion of B ε (k) from the one-particle case (cf. [2, Chapter I.5, Eq. (5.56)]), the proof of (4.25) now follows the line of arguments from that case. For the convenience of the reader, we spell out the details in the following.
To start with, we derive from (4.26) the equation The expansion of R ε holds uniformly in P with Q = Q(P ) ≤ K. To obtain an expression for the inverse on the right side of (4.27), we now apply the identity (4.17). This yields that as operators in L (L 2 (R 2 )). By the definitions of β(Q) and R ε , and by 2πa = u | v , and where R ε = O(| ln(ε)| −1 ) was used. Note that the sum of (4.29) and (4.30) is of order 1/| ln(ε)| because u | v cancels. It follows that the second term in (4.28) is of order | ln(ε)|. Hence we may ignore the first summand, (1+R ε ) −1 , in Equation (4.28) and we may replace the numerator in the second summand by |u v|. It follows that, uniformly in P with Q = Q(P ) ≤ K, with α defined by (4.23). Finally, it follows from (4.27) and from the asymptotics of g ε with a = u | v /(2π) that (4.25) with D(z, P ) from (4.22) holds true for all sufficiently small ε > 0. This proves the lemma.
For later convenience, we now collect the conclusions of this section. To this end, we need to reinstall the index σ, which we dropped at the beginning of the section. Let σ ∈ I be a fixed pair. If V σ ∈ L 1 ∩ L 2 (R 2 ) and |r| 2s |V σ (r)| dr < ∞ for some s ∈ (0, 2), then the analogues of Lemma 4.2 and Proposition 4. (4.22) and (4.23). For general pairs σ = (i, j), the operator Θ(z) σσ acts pointwise in

Analysis of Λ ε (z) off
If N > 2, then Λ ε (z) has an off-diagonal part Λ ε (z) off defined by Eq. (4.4). We will see in this section that Λ ε (z) off is uniformly bounded in ε > 0 and z > 0 and that, upon introducing a cutoff, it has a limit as ε ↓ 0. These results will allow us in Section 4.3 to prove existence of lim ε→0 Λ ε (z) −1 for sufficiently large z > 0. With this goal in mind, the results of this section are formulated for z > 0 only, although most of them still hold in a slightly modified way for general z ∈ ρ(H 0 ).

Uniform boundedness of
Recall from Eqs. (4.4) and (4.1) that the non-vanishing components of Λ ε (z) off are given by We will prove in this section, among other things, that for σ = ν the norm Λ ε (z) σν is uniformly bounded in ε > 0 and z > 0, the main result being Proposition 4.4 below. The presence of the distinct coordinate transformations K σ and K ν in (4.35) makes the proof very technical and somewhat tedious. Since the tools used in this proof are not needed anymore in the sequel, it is possible and advisable to take Proposition 4.4 for granted and to skip the proof at first reading.
With this said, we now start developing the tools for proving Proposition 4.4. To estimate the norm of Λ ε (z) σν , we compute its integral kernel in terms of the Green's function G z,m of H 0 , where the vector m := (m 1 , ..., m N ) collects the masses of the N particles. By a simple scaling argument, where G 2N z denotes the usual Green's function of −∆+z : As the choice of the pair σ is immaterial for our estimates, we can assume that σ = (1, 2) and ν = (k, l) = (1, 2). Now, with U ε , K (1,2) and K * (k,l) defined by (2.5), (2.3) and (2.4), respectively, we find that (4.35) defines an integral operator and, for ψ ∈ X (k,l) , The second equation of (4.37) was obtained by the substitution and two more integrations were introduced that are compensated by delta distributions. In the following, we distinguish two cases: In the first case σ = (1, 2) and ν = (k, l) have one particle in common, so k ∈ {1, 2} and l ≥ 3. In the second case σ and ν are composed of distinct particles, which means that 3 ≤ k < l ≤ N . If σ = (1, 2) and ν = (1, l) with l ≥ 3, then, after the evaluation of the delta distributions in x ′ 1 and x ′ l , (4.37) shows that the operator Λ ε (z) σν simply acts by convolution in (x 3 , ... x l ..., x N ). Consequently, the explicit formula (4.36) for G z,m and Lemma A.1 (vi) imply that Λ ε (z) σν acts pointwise in the conjugate variable p (1,l) := (p 3 , ... p l ..., p N ) by the operator Λ ε (z, p (1,l) ) σν that has the kernel where Similar considerations for σ = (1, 2) and ν = (2, l) with l ≥ 3 show that Λ ε (z) σν acts pointwise in p (2,l) := (p 3 , ... p l ..., p N ) by the operator Λ ε (z, p (2,l) ) σν with kernel where By inspection, the kernels (4.38) and (4.39) only differ by the permutations (2,l) and the reflection r → −r, which will allow us to analyze them simultaneously.
So far, we have considered all operators that appear in the case of N ≤ 3 particles. Let now N > 3, σ = (1, 2) and ν = (k, l) with 3 ≤ k < l ≤ N . Then, after the evaluation of the delta distributions in x ′ k and x ′ l , it follows from (4.37) that Λ ε (z) σν acts by convolution in (x 3 , ... x k ... x l ..., x N ). Hence, the action of Λ ε (z) σν is pointwise in the conjugate variables Besides the bound for the norm of Λ ε (z) σν , we shall also estimate the error caused by cutting off the potential outside some ball of radius h > 0 in the following proposition. This, in turn, will reduce the proof of our convergence result (see Proposition 4.8) to the case of compactly supported potentials. For any pair σ ∈ I and h > 0, let and let Λ h ε (z) σν denote the operator Λ ε (z) σν , where the potentials V σ and V ν are replaced by V h σ and V h ν , respectively. This means that the kernel of Λ h ε (z) σν emerges from the kernel of Λ ε (z) σν by replacing u σ and v ν with u h σ := sgn(V σ )|V h σ | 1/2 and v h ν := |V h ν | 1/2 , respectively.
Proof. Without loss of generality, we may assume that σ = (1, 2) and then we have to establish (4.41) and (4.42) for all pairs ν = (1, 2). We first consider the case ν = (1, l) with l ≥ 3. The proofs of (4.41) and (4.42) are similar: It follows from (4.38) that in both cases we have to estimate the norm of an operator that, for fixed p (1,l) , is given by a kernel of the form where X ε := X (1,2)(1,l),ε and Q := Q (1,l) for short. Explicitly, we have W (r, r ′ ) = u (1,2) (r)v (1,l) (r ′ ) in the case of (4.41) and W (r, r ′ ) = u (1,2) in the case of (4.42). Therefore, we only demonstrate the desired estimate in the case of (4.41).
For ψ ∈ L 2 (R 2 × R 2 × R 2 ), the Cauchy-Schwarz inequality in the r ′ -integration yields For a further estimate of (4.43), we fix r ∈ R 2 . Then triangle inequality and the sequence of 12 r → x ′ 2 in the first step and the monotonicity of the Green's function w.r.t. z and m 1 , m 2 , m l (cf. Lemma A.1 (v)) in the second step yield that By Lemma A.3, F is bounded and F ≤ (4 √ 2m 2 ) −1 . Using this together with the fact that ψ = ψ , we obtain from (4.45) that where the right side is independent of r ∈ R 2 and p (1,l) . Hence, (4.41) for σ = (1, 2) and ν = (1, l) now follows by combining (4.43), (4.44) and (4.46).
As already mentioned, the bound (4.42) is obtained in the same manner with the only difference that W (r, r ′ ) = u σ (r)v ν (r ′ ) − u h σ (r)v h ν (r ′ ) in this case. Then, (4.44) has to be replaced by the identity This completes the proof of (4.41) and (4.42) if σ = (1, 2) and ν = (1, l) with l ≥ 3.
If σ = (1, 2) and ν = (k, l) with 3 ≤ k < l ≤ N , then the operator Λ ε (z) σν acts pointwise in p (k,l) = (p 3 , ... p k ... p l ..., p N ) by the operator Λ ε (z, p (k,l) ) σν that is defined by the kernel (4.40). In this case, the above estimates have to be slightly adjusted. The role of the operator F is now played by the operator B ∈ L L 2 (R 2 × R 2 × R 2 ) from Lemma A.3. As the bounds for F and for B in Lemma A.3 differ by a factor of m −1 , while there is one mass factor more in front of the Green's function for ν = (k, l) with 3 ≤ k < l ≤ N than for ν = (1, l) with l ≥ 3, we again obtain (4.41) and (4.42) with C(σ, ν) given in the statement of the proposition.

The off-diagonal limit operators
This section is a preparation for the next one, where we shall be concerned with the convergence, as ε → 0, of From Proposition 3.3 we know that in L (H 2 (R 2N ), X σ ). Here, |u σ : X σ → X σ is defined by |u σ ψ = u σ ⊗ ψ and v ν | : X ν → X ν is the adjoint of |v ν . The negative of the formal composition of the limits in (4.48) and (4.49) is the operator In the remainder of this section, we show that (4.50) defines an element Θ(z) σν ∈ L (X ν , X σ ). We begin by computing representations of T σ and G(z) * ν in Fourier space. Using the definitions (3.2), (3.3) and (2.3) of T σ , τ and K σ , respectively, we find for σ = (i, j) and ψ ∈ D(T σ ) that ( T (i,j) ψ) (P, p 1 , ... p i ... p j ..., p N  To compute the Fourier transform of G(z) * ν w for z > 0, ν = (k, l), and w ∈ X ν , we use G(z) ν = T ν R 0 (z) in combination with (4.51) and the substitution After a straightforward computation, we find that w | G(z) ν ψ = G(z) * ν w | ψ for any ψ ∈ H , where + p l , p 1 , ... p k ... p l ..., p N ) . In the proof of Proposition 4.5 we will need the following result taken from [6, Lemma 3.1]. For the convenience of the reader, we give a short proof with a different constant here. (4.53) Proof. Let K denote the integral operator in L 2 (R 2 ) that is defined by the kernel K(x; x ′ ) = (|x| 2 + |x ′ | 2 ) −1 . Using the Schur test with h(x) = |x| −1 , it is straightforward to verify that K defines a bounded operator and that K ≤ ess sup Hence, | f | , K | g | ≤ π 2 f g , which establishes (4.53).
With the help of Lemma 4.7 we can now prove convergence of the regularized operators (1 ⊗ χ σν,c )Λ ε (z) σν for pairs σ = (i, j) = (k, l) = ν, and a cutoff function χ σν,c defined by Proof. Assume, for the moment, that for all h > 0, where Λ h ε (z) σν was introduced in Section 4.2.1 and Λ h (z) σν := |u h σ v h ν |⊗Θ(z) σν . By Proposition 4.5, Θ(z) σν = −T σ G(z) * ν defines a bounded operator and Θ(z) σν ≤ C(σ, ν), so the definition (4.64) and the corresponding identity for Λ h (z) σν imply that which is the equivalent of (4.42) for ε = 0. Now, the general case of (4.63), where u σ and v ν do not have compact support, follows from (4.65), (4.42) and (4.66) by a simple δ/3-argument. It remains to prove (4.65). As the choice of the pair ν is immaterial for the following arguments, it suffices to consider the case ν = (1, 2) only. Moreover, we may assume that Next, setting for any n ∈ N 0 . Hence, we conclude from (4.68) and (4.69) that To complete the proof of (4.65), we have to move the cutoff to the left on both sides of (4.70). We start by considering the limit operators. For a given ψ ∈ X ν , let ψ := S(z) * ν ψ. Then (4.69) shows that χ ν,c ψ ∈ H n (R 2N ) for any n ∈ N, so it follows from a standard Sobolev embedding theorem (see e.g. [15,Theorem 8.8]) that χ ν,c ψ defines a smooth function. In particular, χ ν,c ψ ∈ D(T σ ) and T σ (χ ν,c ψ) is explicitly given by (1.6). Now, a direct calculation, using the defining relations for χ ν,c and χ σν,c , yields that where, by (3.6) and Proposition 4.
so the limit operator in (4.70) agrees with the desired limit operator in (4.65). It remains to show that the left side of (4.70) agrees with the left side of (4.65) with Λ h ε (z) σν given by (4.67). If σ = (k, l) with 3 ≤ k < l ≤ N , then this follows immediately from U ε K σ χ ν,c = (1 ⊗ χ σν,c )U ε K σ . If σ = (1, l) with l ≥ 3, then, by some abuse of notation, For the purpose of computing the limit in (4.70), the right side of (4.71) may be replaced by χ c (|x 2 − R|)U ε K σ because χ c is Lipschitz, u σ has compact support and, by Proposition 4.4, Λ h ε (z) σν is uniformly bounded in ε > 0. Again, (4.65) follows. The remaining case σ = (2, l) with l ≥ 3 is treated similarly.
Proposition 4.5 implies that Θ(z) off ∈ L (X), while, by (4.32), Θ(z) diag is unbounded in X. With the help of Π and U , Equation (4.31) can now be written as Here, the operator product A • B is defined by (A • B) ij = A ij ⊗ B ij , a notation inspired by the Hadamard-Schur product of matrices.
The following proposition proves (4.72) in these new notations: Proposition 4.9. There exist ε 0 , z 0 > 0 such that the operator matrix Λ ε (z) is invertible in X for all ε ∈ (0, ε 0 ) and z > z 0 , and then where Θ(z) is a closed and invertible operator in X.
To simplify the right side of (4.85), we first note that the inverse (1 + L(z)) −1 is only needed on Ran Π * , which, in view of (4.84), is left invariant by L(z). Explicitly, we have that Secondly, by Proposition 4.8, we have the factorization property where Θ(z) σν ∈ L (X ν , X σ ) for σ = ν. From (4.87), it follows that where the components of U = ( U σν ) σ,ν∈J are defined by and the identity U σν = U σλ |u λ v ν | was used. Furthermore, a direct computation, using the identity ( U • A)( U • B) = ( U • (AB)), shows that, on Ran Π, where the last equation follows from the second resolvent identity and the second to last equation used U σλ U λν = U σν .
Since J ν = sgn(V ν ) is bounded and since, by Proposition 3.3, lim ε→0 A ε,ν R 0 (z) = S(z) ν exists for all ν, we conclude that we can take the limit ε → 0 on the right side of (5.1). We find that Expression (5.2) can be simplified as follows: By Proposition 3.3, we have that where G(z) ∈ L (H , X) is given by G(z)ψ = (G(z) σ ψ) σ∈J . This expression for R(z) agrees with the right side of (1.7). Moreover, it also shows that R(z) only depends on V σ (σ ∈ J ) via the parameters β σ that are defined by (4.33).
To prove Theorem 1.1 for z > z 0 , it remains to show that R(z) indeed defines the resolvent (H + z) −1 of a self-adjoint operator H. This follows from the Trotter-Kato Theorem (see e.g. [7,Theorem 5] and [21,Theorem VIII.22] . For compactly supported, smooth functions ψ, T σ ψ agrees with (1.6), so Ker T contains the dense set denotes the union of the contact planes of the N particles. Let φ ∈ (Ran R(z)) ⊥ . Then, for any ψ ∈ Ker T , it follows from (5.3) and G(z) = T R 0 (z) that As Ker T ⊂ H is dense, this implies that φ = 0, i.e. Ran R(z) ⊂ H is dense. Now, by the Trotter-Kato Theorem, there exists a self-adjoint operator H such that R(z) = (H + z) −1 † In [21] existence of limε→0(Hε + z) −1 in two points z with ± Im(z) > 0 is assumed, but the proof can be adapted to the case where the limit exists for all z from a non-empty open interval.
To prove (i), we use the fact that Θ(z) does not depend on the particular choices of V σ , a σ and b σ as long as the parameters β σ and the set J remain unchanged. For σ ∈ J we choose V σ > 0, a σ = V σ (r) dr/(2π) and b σ to solve (4.33) and (4.34) for the given values of β σ . For σ ∈ I \ J we set V σ = 0. Moreover, we choose z ∈ (z 0 , ∞) so that (4.74) holds. Then u σ = v σ for all pairs σ ∈ I and hence Λ ε (z) is self-adjoint for all ε > 0. Now, it follows from (4.74), or more directly from (4.72), that Θ(z) −1 is self-adjoint, too. Therefore, Θ(z) is self-adjoint for z ∈ (z 0 , ∞) and (i) for general z ∈ ρ(H 0 ) now easily follows from (ii).

The quadratic form of the Hamiltonian
In this section we determine the quadratic form of H and we show, in the case of N particles of mass one, that it agrees with a quadratic form introduced in [ where w = (w σ ) σ∈J ∈ D(Θ(z)) is uniquely determined by (1.8) and (1.9).
Next, we are going to determine an explicit description of the closure of the quadratic form from Lemma 6.1. From (4.32), it follows that Θ(z) σσ ≥ c 1 ln(z) + c 2 in operator sense, where c 1 > 0 and c 2 ∈ R depend on the pair σ ∈ J but not on z ∈ (0, ∞). Combining this with the fact that Θ(z) off is uniformly bounded in z ∈ (0, ∞), we see that Θ(z) ≥ c > 0 for sufficiently large z ∈ ρ(H) ∩ (0, ∞). For such z, we introduce a quadratic form q with domain which agrees with the right side of (6.1) if ψ ∈ D(H) ⊂ D(q). Similarly to [6, Lemma 3.2], one verifies that G(z) * w / ∈ H 1 (R 2N ) for any w ∈ D(Θ(z) 1/2 ) \ {0}, so w = w ψ is uniquely determined by ψ ∈ D(q) and, in particular, q is well-defined. Furthermore, it is not hard to see that q is bounded from below and closed. Hence, there exists a unique self-adjoint operator H q associated with q. We are going to show that H q = H. Let ψ ∈ D(H q ) be fixed. Then, by the definitions of q and H q , for all φ ∈ D(q), Choosing w φ = 0, wee see that any φ ∈ H 1 (R 2N ) belongs to D(q), and hence, for φ ∈ H 1 (R 2N ), Eq. (6.4) becomes This equation shows that ψ 0 := ψ − G(z) * w ψ ∈ D(H 0 ), and that We have thus verified condition (1.8) of Corollary 1.2. It remains to check condition (1.9). Given w ∈ D(Θ(z) 1/2 ), we choose φ := G(z) * w, so that φ − G(z) * w = 0 ∈ H 1 (R 2N ). It follows that φ ∈ D(q), and from Eqs. (6.4) and (6.6), we see that, for all w ∈ D(Θ(z) 1/2 ), This equation implies that w ψ ∈ D(Θ(z)) and that Θ(z)w ψ = T ψ 0 , which is condition (1.9). Corollary 1.2 now shows that ψ ∈ D(H) and, in view of Eq. (6.6), that H q ⊂ H. From the self-adjointness of H q and H, we conclude that H q = H.
In the proof of Proposition 4.4 two integral operators F and B depending on G d z were introduced. We close this section with the bypassed proof of their boundedness: Lemma A.3. Let z, m > 0. Then the operator F : L 2 (R 2 × R 2 ) → L 2 (R 2 × R 2 ) that is defined in terms of the kernel Proof. The bounds for F and B are both based on the Schur test. In the case of F , the Schur test, in a general version, yields that for any measurable test function h : R 2 × R 2 → (0, ∞), provided that the right side is finite. We choose the test function h(x, y) := |x − y| −1 , so after scaling we may assume that m = 1.
To evaluate the right side of (A.3), we insert the integral representation (A.1) of the Green's function G 6 z and we substitute x − y ′ → y ′ . This results in the estimate ess sup Using a rearrangement inequality (see [15,Theorem 3.4]), the last integral is bounded by For the remaining integral, we use (x − x ′ ) 2 + (y − x ′ ) 2 = 1 2 (2x ′ − x − y) 2 + (x − y) 2 together with the substitution 2x ′ − x − y → 2x ′ . This yields that π ess sup In the last equality, the ess sup cancels after the scaling t → (x − y) 2 t/8. The remaining integral can be explicitly computed with the help of the substitution t := t −1 . We find that where Γ denotes the Gamma function. Using (A.5), (A.6) and (A.7) to bound the right side of (A.4), the Schur test yields that F is bounded and its norm satisfies the desired estimate. The proof in case of the operator B is similar, but for the convenience of the reader we go into details here. First, applying the Schur test with the test function h(y, w) = |y − w| −1 gives B ≤ ess sup x,y,w |y − w| dx ′ dy ′ dw ′ |y ′ − w ′ | −1 K B (x, y, w; x ′ , y ′ , w ′ ) , provided that the right side is finite. After scaling, we may again assume that m = 1. Next, inserting the integral formula (A.1) for G 8 z and substituting x − y ′ → y ′ , it follows that Similarly to (A.5), a rearrangement inequality now yields that In the remaining integral, we use (y − x ′ ) 2 + (w − x ′ ) 2 = 1 2 (2x ′ − w − y) 2 + (y − w) 2 in combination with the substitutions 2x ′ − w − y → 2x ′ and x − w ′ → w ′ . This leads to π ess sup

B Konno-Kuroda formula
In this section we sketch the proof of the Konno-Kuroda resolvent identity, see [13,Eq. (2.3)], for operators of the type (B.1). The main difference between Theorem B.1 below and [13] is that we do not assume that φ(z) defined by (B.2) extends to a compact operator for some (and hence all) z ∈ ρ(H 0 ).
Let H and X be arbitrary complex Hilbert spaces, let H 0 ≥ 0 be a self-adjoint operator in H and let A : D(A) ⊂ H → X be densely defined and closed with D(A) ⊃ D(H 0 ). Let J ∈ L ( X) be a self-adjoint isometry and let B = JA. Suppose that BD(H 0 ) ⊂ D(A * ) and that A * A and A * B are H 0 -bounded with relative bound less than one. Then Proof.
Step 1. AR 0 (c) 1/2 is bounded for c > 0, and A(H + c) −1/2 is bounded for c > 0 large enough. As A * A is H 0 -bounded with relative bound less than one, the operator H 0 − A * A is bounded from below. This implies that, for all ψ ∈ D(H 0 ) and all c > 0, This easily follows from Step 1 and from the first resolvent identity.
Assume that X = I i=1 X i , where I ∈ N and X i are complex Hilbert spaces for i = 1, ..., I. Then we consider operators of the more general form Hence, Theorem B.1 implies the following corollary: