Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three

We study the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan--Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The analysis is based on the construction of a suitable quadratic form which is shown to be closed and bounded from below. Then, domain and action of the corresponding Hamiltonian are completely characterized and a regularity result for the elements of the domain is given. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non local interactions, also called separable potentials, with a suitably renormalized coupling constant.


Introduction
In a system of nonrelativistic quantum particles at low temperature, the thermal wavelength is typically much larger than the range of the two-body interactions and therefore the details of the interactions are irrelevant. In these conditions, the effective behavior of the system is well described by a Hamiltonian with zero-range forces, where the only physical parameter characterizing the interaction is the scattering length. The mathematical construction of such Hamiltonians as self-adjoint and, possibly, lower-bounded operators is straightforward in dimension one since standard perturbation theory of quadratic forms can be used. Moreover, the Hamiltonian can be obtained as the resolvent limit of approximating Hamiltonians with rescaled two-body smooth potentials (see [4] for the case of three particles and [18] for n bosons). On the contrary, in dimensions two and three the interaction is too singular and more refined techniques are required for the construction. The two-dimensional case is well understood ( [11,12], see also [19] for applications to the Fermi polaron model), and it has been recently shown [17] that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled smooth potentials and with a suitably renormalized coupling constant. In dimension three, the problem is more subtle due to the fact that a natural construction in the case of n 3 particles, obtained following the analogy with the one particle case, leads to the so-called TMS Hamiltonian [35] which is symmetric but not self-adjoint. Furthermore, all its self-adjoint extensions are unbounded from below. Such instability property, known as Thomas effect, can be seen as a fall to the center phenomenon, and it is due to the fact that the interaction becomes too strong and attractive when (at least) three particles are very close to each other. This phenomenon The research that led to the present paper was partially supported by a grant of the group GNFM of INdAM . does not occur in dimension two because the singularity of the wave function at the coincidence hyperplane is a mild logarithmic one. The Thomas effect was first noted by Danilov [10] and then rigorously analyzed by Minlos and Faddeev [28,29], and it makes the Hamiltonian unsatisfactory from the physical point of view (for some recent mathematical contributions see, e.g., [5,6,14,22] with references therein). For other approaches to the construction of many-body contact interactions in R 3 , we refer to [2,31,36]. We note that a different situation occurs in the case (which is not considered here) of a system made of two species of fermions interacting via zero-range forces, where it happens that for certain regime of the mass ratio the TMS Hamiltonian is in fact self-adjoint and bounded from below (for mathematical results in this direction see, e.g., [7,8,9,15,24,25,26,27,30,32]). For a general mathematical approach to the construction of singularly perturbed self-adjoint operators in a Hilbert space, we refer to [33,34]. Inspired by a suggestion contained in [28], in this paper we propose a regularized version of the TMS Hamiltonian for a system of three bosons and we prove that it is self-adjoint and bounded from below. Furthermore, we show that the Hamiltonian is the norm resolvent limit of approximating Hamiltonians with rescaled non-local interactions, also called separable potentials, and with a suitably renormalized coupling constant. We stress that a more interesting problem from the physical point of view would be the approximation in norm resolvent sense by a sequence of Hamiltonians with (local) rescaled potentials as in [17] for the two dimensional case and [4,18] for the one-dimensional case. Such a result is more difficult to prove, and we plan to approach it in a forthcoming work. We also believe that our approach and results can be generalized to the case of three different particles. For the case of a system made of N bosons in interaction with another particle, see [13]. In the rest of this section we introduce our Hamiltonian at heuristic level and discuss some of its properties. Let us consider a system of three identical bosons with masses 1/2 in the center of mass reference frame and let x 1 , x 2 and x 3 = −x 1 − x 2 be the Cartesian coordinates of the particles. Let us introduce the Jacobi coordinates r 23 ≡ x, r 1 ≡ y, namely The other two pairs of Jacobi coordinates in position space are r 31 = − 1 2 x + y, r 2 = − 3 4 x − 1 2 y and r 12 = − 1 2 x − y, r 3 = 3 4 x − 1 2 y. Due to the symmetry constraint, the Hilbert space of states is L 2 sym (R 6 ) = ψ ∈ L 2 (R 6 ) s.t. ψ(x, y) = ψ(−x, y) = ψ Notice that the symmetry conditions in (1.1) correspond to the exchange of particles 2, 3 and 3, 1 and they also imply the symmetry under the exchange of particles 1, 2, i.e., ψ(x, y) = ψ 1 2 x−y, − 3 4 x− 1 2 y . The formal Hamiltonian describing the three boson system in the Jacobi coordinates reads H 0 + µδ(x) + µδ(y − x/2) + µδ(y + x/2) (1.2) where µ ∈ R is a coupling constant and H 0 is the free Hamiltonian Our aim is to construct a rigorous version of (1.2) as a self-adjoint, and possibly bounded from below, operator in L 2 sym (R 6 ). In other words, we want to define a self-adjoint perturbation of the free Hamiltonian (1.3) supported by the coincidence hyperplanes π 23 = {x 2 = x 3 } = {x = 0}, π 31 = {x 3 = x 1 } = {y = x/2}, π 12 = {x 1 = x 2 } = {y = −x/2}.
Following the analogy with the one particle case [1], a natural attempt is to define the TMS operator acting as the free Hamiltonian outside the hyperplanes and characterized by a (singular) boundary condition on each hyperplane. Specifically, on π 23 one imposes ψ(x, y) = ξ(y) x + β ξ(y) + o (1), for x → 0 and y = 0 (1.4) where x := |x|, ξ is a function depending on ψ and a := −β −1 ∈ R (1.5) has the physical meaning of two-body scattering length (and it can be related to µ via a renormalization procedure). Notice that, due to the symmetry constraint, (1.4) implies the analogous boundary conditions on π 31 and π 12 .
As already recalled, the TMS operator defined in this way is symmetric but not self-adjoint and its self-adjoint extensions are all unbounded from below. Therefore, the natural problem arises of figuring out if and how one can modify the boundary condition (1.4) to obtain a bounded from below Hamiltonian. In a comment on this point, at the end of the paper [28] the authors claim that it is possible to find another physically reasonable realization ofH as self-adjoint and bounded from below operator. They also affirm that the recipe consists in the replacement β ξ(y) → β ξ(y) + (Kξ)(y) (1.6) in the boundary condition (1.4), where K is a convolution operator in the Fourier space with a kernel K(p − p ′ ) satisfying with p = |p| and the positive constant γ sufficiently large. The authors do not explain the reason of their assertion neither they clarify the physical meaning of the boundary condition (1.6). They only conclude: "A detailed development of this point of view is not presented here because of lack of space" and, strangely enough, their idea has never been developed in the literature. Almost 20 years later, Albeverio, Høegh-Krohn and Wu [3] have proposed an apparently different recipe to obtain a bounded from below Hamiltonian, i.e., the replacement with y = |y|, in the boundary condition (1.4), where again the positive constant γ is chosen sufficiently large. Also, the proof of this statement has been postponed to a forthcoming paper which has never been published. Even if it has not been explicitly noted by the authors of [3], it is immediate to realize that the two proposals contained in [28] and [3] essentially coincide in the sense that in [28] the term added in the boundary condition is the Fourier transform of the term added in [3]. It is also important to stress that, according to the claim in [28], only the asymptotic behavior of K(p) for |p| → ∞ (see (1.7)) is relevant to obtain a lower-bounded Hamiltonian. Correspondingly, it must be sufficient to require only the asymptotic behavior γy −1 + O(1) for y → 0 for the boundary condition in position space in [3].
The above considerations suggest to define our formal regularized TMS HamiltonianH reg as an operator in L 2 sym (R 6 ) acting as the free Hamiltonian outside the hyperplanes and characterized by the following boundary condition on π 23 where Γ reg is defined by and θ is a real cutoff function. Due to the symmetry constraint, (1.8) implies the boundary condition on π 31 and π 12 We assume different hypothesis on θ depending on the situation. The first possible hypothesis is The simplest choice satisfying (H1) is the characteristic function The second possible hypothesis requires some minimal smoothness Examples satisfying (H2) are θ(r) = e −r/b or θ ∈ C ∞ 0 (R + ) such that θ(r) = 1 for r b, b > 0. We stress that (H2) implies (H1). Hypothesis (H2) will be used only in Sect. 6, where we study the approximation with separable potentials, and in Appendix A.2. Note that the crucial point is the behavior of θ at the origin, which represents the minimal requirement for the regularization of the dynamics at short distances. The support of the function θ is not relevant, in particular a simple choice would be θ(r) = 1. Needless to say, the operatorH reg is only formally defined since its domain and action are not clearly specified. Our aim is to construct an operator which represents the rigorous counterpart ofH reg using a quadratic form method. The main idea of the construction has been announced and outlined in [14], where a more detailed historical account of the problem is given. We also mention the recent paper [22], where the construction is approached using the theory of self-adjoint extensions. Let us make some comments on the formal operatorH reg . As we already remarked, the singular behavior of (Γ reg ξ)(y) for y → 0 corresponds in the Fourier space to a convolution operator whose kernel has the asymptotic behavior (1.7). It will be clear in the course of the proofs in Sect. 3 that such a behavior is chosen in order to compensate the singular behavior of the off-diagonal term appearing in the quadratic form. In this sense, one can say that the singularity of (Γ reg ξ)(y) for y → 0 is the minimal one required to obtain a self-adjoint and bounded from below Hamiltonian.
Concerning the physical meaning of our regularization, we recall that we have replaced the parameter β in (1.4) with Γ reg in (1.8). By analogy with the definition (1.5), we can introduce an effective, position-dependent scattering length which can be interpreted as follows. For simplicity, let us fix β > 0 and choose the cutoff (1.10).
Consider the zero-range interaction between the particles 2, 3 which takes place when x 2 = x 3 , i.e., for x = 0. In these conditions, the coordinate y is the distance between the third particle 1 and the common position of particles 2, 3. Then one has i.e., the effective scattering length associated with the interaction of particles 2, 3 is equal to a if the third particle 1 is at a distance larger than b while for distance smaller than b the scattering length depends on the position of the particle 1 and it decreases to zero, i.e., the interaction vanishes, when the distance goes to zero. In other words, we introduce a three-body interaction which is a common procedure in certain low-energy approximations in nuclear physics. Such three-body interaction reduces to zero the two-body interaction when the third particle approaches the common position of the first two. This is precisely the mechanism that prevents in our model the fall to the center phenomenon, i.e., the Thomas effect. The paper is organized as follows.
In Sect. 2, starting from the formal HamiltonianH reg , we construct a quadratic form which is the initial point of our analysis and we formulate our main results. In Sect. 3, we prove that the quadratic form is closed and bounded from below for any γ larger than a threshold explicitly given. In Sect. 4, we characterize the self-adjoint and bounded from below Hamiltonian H uniquely associated with the quadratic form which is the rigorous counterpart ofH reg . In Sect. 5, we introduce a sequence of approximating Hamiltonians H ε with rescaled separable potentials and a renormalized coupling constant and we prove a uniform lower bound on the spectrum. In Sect. 6, we show that the Hamiltonian H is the norm resolvent limit of the sequence of approximating Hamiltonians H ε .
In the Appendix, we prove a technical regularity result for the elements of the domain of H.
In conclusion, we collect here some of the notation frequently used throughout the paper. x is a vector in R 3 and x = |x|.
-For a linear operator A acting in position space, we denote byÂ = F AF −1 the corresponding operator in the Fourier space.
-H s (R n ) denotes the standard Sobolev space of order s > 0 in R n .
-· and (·, ·) are the norm and the scalar product in L 2 (R n ), · L p is the norm in L p (R n ), with p = 2, and · H s is the norm in H s (R n ). It will be clear from the context if n = 3 or n = 6.
-f π ij ∈ H s (R 3 ) is the trace of f ∈ H 3/2+s (R 6 ), for any s > 0. -c will denote numerical constant whose value may change from line to line.

Construction of the Quadratic Form and Main Results
Here, we describe a heuristic procedure to construct the quadratic form (ψ,H reg ψ) associated with the formal HamiltonianH reg defined in the introduction. Since we mainly work in the Fourier space, we introduce the coordinates k 23 ≡ k, k 1 ≡ p, conjugate variables of x, y where p 1 , p 2 and p 3 = −p 1 − p 2 are the momenta of the particles. The other two pairs of Jacobi coordinates in momentum space are In the Fourier space, the Hilbert space of states is equivalently written as The symmetry conditions in (2.1) correspond to the exchange of particles 2, 3 and 3, 1 and they also imply the symmetry under the exchange of particles 1, 2, i.e., ψ(k, p) =ψ 1 2 k − 3 4 p, −k − 1 2 p . Moreover the free Hamiltonian isĤ We also introduce the "potential" produced by the "charge density" ξ distributed on the hyperplane π 23 by and one can verify that the function G λ 23 ξ satisfies the equation in distributional sense. Analogously, we have and the potential produced by the three charge densities is Note that the function G λξ is symmetric under the exchange of particles; hence, it belongs to L 2 sym (R 6 ), see Eq. (2.1).
(2.12) By (2.11), (2.12) and the boundary condition (2.10), we finally arrive at the definition of the following quadratic form where (Γ regξ )(p) is the Fourier transform of the function defined in Eq. (1.9). We define the quadratic form F on the domain From the explicit expression of the potential (2.6), one immediately sees that for any In the rest of the paper, we assume Definition 2.1 as the starting point of our rigorous analysis. Let us conclude this section collecting the main results we prove in the paper. First, we show that for any γ > γ c , where the quadratic form F on the domain D(F ) is closed and bounded from below. This is the content of the next theorem whose proof is presented in Sect. 3.

18)
The proof of Theorem 2.4 is deferred to Sect. 4. The next question we address is the approximation through a regularized Hamiltonian H ε , D(H ε ) with non-local interactions, also known as separable potentials. In order to define the approximating model, we need to first introduce some notation.
, spherically symmetric, real valued, nonnegative and such that dx χ(x) = 1. Moreover, set For all ε > 0, we define the scaled function χ ε as and the operator g ε on L 2 (R 3 ) with Γ reg given in (1.9). Then, the approximating Hamiltonian H ε , D(H ε ) on L 2 sym (R 6 ) is defined as where S is the permutation operator exchanging the triple of labels (1, 2, 3) in the triple (2, 3, 1). So that, Taking into account that χ ε → δ for ε → 0 in distributional sense, one sees that the three interaction terms in (2.23) for ε → 0 formally converge to zero-range interactions supported on the hyperplanes π 23 , π 31 , π 12 . Moreover, the operator g ε plays the role of renormalized coupling constant. We also note that in position space g ε reduces to the multiplication operator by g ε (y) which for ε small behaves as In particular, if we assume for simplicity that θ is the characteristic function (1.10) then we find that for y > b we have the standard behavior required to approximate a point interaction in dimension three with scattering length −β −1 (see [1], chapter II.1.1, pages 111-112), while for y b we have introduced a dependence on the position y such that the modified scattering length − β + γ y −1 goes to zero as y → 0. In Sect. 5 (see Theorem 5.7), we prove a uniform lower bound for the spectrum of H ε , i.e., we show that there exists λ 1 > 0, independent of ε, such that inf σ(H ε ) > −λ 1 . Finally, we prove the norm resolvent convergence of H ε , D(H ε ) to H, D(H) as ε → 0. More precisely, the following result holds true and it is proved in Sect. 6.

Analysis of the Quadratic Form
In this section, we prove closure and boundedness of the quadratic form F defined by (2.13)-(2.16) for γ > γ c with γ c defined in (2.17). To this end we first study the quadratic form Φ λ in L 2 (R 3 ) given by (2.15) and acting on the domain D(Φ λ ) = H 1/2 (R 3 ). Recalling the definition of Γ reg given in (1.9) and using the fact that Note that a ∈ L ∞ (R 3 ) if we assume (H1) and a, ∇a ∈ L ∞ (R 3 ) if we choose (H2). We will show that Φ λ is equivalent to the H 1/2 -norm. First, we prove that Φ λ (ξ) can be bounded from above by ξ 2 H 1/2 . This is the content of the next proposition which ensures that Φ λ is well defined on D(Φ λ ) = H 1/2 (R 3 ).
The next step is to bound from below Φ λ , which is our main technical result for the construction of the Hamiltonian. Our main tool is the decomposition of the functionξ into partial waves (for an alternative approach see, e.g., [30]). Then, we writê where Y ℓ m denotes the Spherical Harmonics of order ℓ, m and p = (p, θ p , ϕ p ) in spherical coordinates. Accordingly, we find the following decomposition of the quadratic form Φ λ where φ λ ℓ is the quadratic form whose action on g ∈ L 2 ((0, +∞), p 2 p 2 + 1dp) is given by (see, e.g., with P ℓ (y) = 1 2 ℓ ℓ! d ℓ dy ℓ (y 2 − 1) ℓ the Legendre polynomial of degree ℓ. In the next lemma, we investigate the sign of φ λ off,ℓ. Lemma 3.2. Let g ∈ L 2 (R + , p 2 p 2 + 1 dp) and λ > 0. Then, Proof. The proof follows [8]. For the sake of completeness, we give the details below. First, we rewrite where in the last line we integrated by parts ℓ times. Next, we note that It's easy to see that B ℓj = 0 if ℓ and j do not have the same parity. Moreover, B ℓj 0 if ℓ, j are even and B ℓj 0 if ℓ, j are odd. Then, (3.8) yields (3.7).
Thanks to Lemma 3.2, in order to obtain a lower bound we can neglect φ λ off,ℓ with ℓ odd and focus on φ 0 off,ℓ which control φ λ off,ℓ with ℓ even. In the next lemma we show that φ 0 off,ℓ and φ reg,ℓ can be diagonalized, note that we also include the analysis of φ 0 off,ℓ with ℓ odd for later convenience.
Lemma 3.3. Let g be a real analytic function whose Taylor expansion near the origin has a radius of convergence bigger or equal than one. Define with P ℓ being the Legendre polynomials. Then for any ℓ ∈ N, we have Proof. Using P ℓ (y) = 1 2 ℓ ℓ! d ℓ dy ℓ (y 2 − 1) ℓ and integrating by parts, one easily prove the first claim. Integrating by parts again, one finds and the monotonicity follows from the first claim.
(3.10) Moreover, For reader's convenience we give the details below.
With the change of variables p 1 = e x 1 and p 2 = e x 2 , we rewrite Taking the Fourier transform we get with (see, e.g., [16, p. 511]) .
In order to prove the monotonicity properties (3.11), it is sufficient to notice that the Taylor expan- have positive coefficients and invoke Lemma 3.3. A dilation of a factor 2 preserve the positivity of the coefficients and then also (3.12) follows from Lemma 3.3.
A key ingredient of the proof of closure and boundedness from below of Φ λ is the following lemma.

(3.19)
The explicit computation of the two integrals in (3.19) yields It remains to show f (k) 0 to conclude. We note that f (k) is even and thus it is enough to consider k 0. We have f (0) > 0, f 2 (k) 0 and For γ > γ c , let us choose s * such that max{0, 1 − π √ 3 (γ −γ c )} < s * < 1. Hence, we also have f 1 (k) 0 and the proof is complete.
We are ready to prove the lower bound for Φ λ (ξ). This is the content of the next proposition which together with Proposition 3.1 shows that Φ λ defines a norm equivalent to · H 1/2 . Proposition 3.6. Assume (H1) and γ > γ c . Then, there exist λ 0 > 0 and c 0 > 0 such that for any λ > λ 0 .
Proof. By (3.5), (3.6) and Lemma 3.2, we get Then, using Lemmata 3.4, 3.5, we obtain To conclude, we note We are now in position to prove Theorem 2.3 formulated in Sect. 2.
Remark 3.7. By (3.21) one has We underline that λ 0 depends on γ both via a L ∞ and via Λ. In particular, as γ → γ c we have s * → 1, so that Λ → 0 and λ 0 → ∞. In the concrete case (1.10), we can take From the proof of Theorem 2.3, it is clear that −λ 0 is a lower bound for the infimum of the spectrum of H.
Remark 3.8. We expect γ c to be optimal that is, if γ < γ c one could argue as in [15] and prove that F is unbounded from below.

Hamiltonian
In this section, we explicitly construct the Hamiltonian of our three bosons system. Let us first consider the quadratic form Φ λ , D(Φ λ ) = H 1/2 (R 3 ) in L 2 (R 3 ) (see (2.15)). As a straightforward consequence of Point (i) of Theorem 2.3, such a quadratic form is closed and positive, and therefore it uniquely defines a positive, self-adjoint operator Γ λ in L 2 (R 3 ) for λ > λ 0 characterized as follows In the appendix, we prove that D(Γ λ ) = H 1 (R 3 ) for γ > γ * c (see Proposition A.2) and Let us now consider the quadratic form F , D(F ) in L 2 sym (R 6 ). By Theorem 2.3, such quadratic form uniquely defines a self-adjoint and bounded from below Hamiltonian H, D(H) in L 2 sym (R 6 ), next we prove Theorem 2.4 which characterizes its domain and action.
Remark 4.1. We emphasize that the Hamiltonian H, D(H) is the rigorous counterpart of the formal regularized TMS Hamiltonian introduced in Sect. 1. Indeed, for any ψ ∈ L 2 sym (R 6 ) ∩ C ∞ 0 (R 6 \ ∪ i<j π ij ) we have ψ ∈ D(H) and Hψ = H 0 ψ, i.e., the Hamiltonian acts as the free Hamiltonian outside the hyperplanes. Moreover, we show that the boundary condition (1.8) is also satisfied. Let us consider ψ ∈ D(H) and let us recall that the corresponding charge ξ belongs to H 1 (R 3 ). For x = 0 we write and we compute the limit of the above expression for x → 0 in the L 2 -sense. Taking into account of (2.7), we have which, by dominated convergence theorem, converges to zero for x → 0. Moreover, for any η ∈ by dominated convergence theorem and the same is true for (G λ 12 ξ)(x, y) − (G λ 12 ξ)(0, y). Note that (G λ 31 ξ)(0, y) + (G λ 12 ξ)(0, y) = −(Γ λ off ξ)(y). For the last term in (4.6) we have and then w λ (x, y) − w λ (0, y) → 0 for x → 0 in L 2 (R 3 ). Taking into account the above estimates, the condition Γ λ ξ = w λ π 23 and the decomposition Γ λ = Γ λ diag + Γ λ off + Γ reg we conclude which is precisely the boundary condition (1.8) satisfied in the L 2 -sense.
Let us characterize the resolvent of our Hamiltonian. We first introduce the shorthand notation for the operator G λ 23 = G λ (see (2.2)), i.e., its adjoint is Next, we prove the following preliminary result.
Proof. We note that So that Let us recall the definition of the operator S given in (2.24), additionally we notice that We note that the second equality is a consequence of the first one. Furthermore, we have S * = S 2 . Taking into account of (2.6), (4.7), (2.24), (4.8), we can write We claim that the resolvent (H + λ) −1 of H, D(H) computed in z = −λ < −λ 0 is given by where R λ 0 = (H 0 + λ) −1 and (Γ λ ) −1 is a well defined and bounded operator in L 2 (R 3 ) since Φ λ is coercive. Indeed, let us consider R λ f for f ∈ L 2 sym (R 6 ). We have

Approximating Hamiltonian
In this section, we prove a uniform bound on the infimum of the spectrum of H ε introduced in Sect. 2 and obtain the Konno-Kuroda formula for its resolvent (Theorem 5.7).
Remark 5.1. Let us recall the scaled function χ ε defined in (2.21), and the definitions of the constants ℓ and ℓ ′ in (2.20). The assumptions on χ imply thatχ is real valued, Lipschitz, that ℓ, ℓ ′ < ∞, and that Moreover, we recall the definition of the infinitesimal, position-dependent coupling constant in (2.22).
In the position space g ε is just the multiplication operator for the function (which we denote by the same symbol) where a(y) was introduced in (3.2). From now on, we always assume that ε < ℓ/(2 a L ∞ ) so that 1 + ε ℓ a(y) + ε ℓ γ y > 1/2 and g ε , as a function, is bounded, in particular g ε L ∞ 8πε/ℓ.
Let us consider the Hamiltonian H ε defined in (2.23). We remark that the term 2 j=0 S j |χ ε χ ε | ⊗ g ε S j * is bounded (although not uniformly in ε) in L 2 (R 6 ), with norm bounded by 3 χ ε 2 g ε L ∞ (24π/ℓ)ε −2 χ 2 , and therefore H ε , D(H ε ) is self-adjoint and bounded from below for any ε > 0. As a first step, we introduce the following operators which will play a crucial role in writing the Konno-Kuroda formula for the resolvent of H ε (see Theorem 5.7).
Lemma 5.4. Let ξ ∈ D(Φ λ ε ), λ > 0 and γ 0 as in (2.20). Then, Proof. By (5.1), the change of variable k = −q − 1 2 p and the action of the operator S (see (2.24)), we find (ξ, Γ λ off,ε ξ) = −8π dp dkχ(εk)ξ(p)χ It is convenient to write the r.h.s. of (5.6) in the position space. To this aim, we denote by R λ 0 (x, y; x ′ , y ′ ) the integral kernel of the operator R λ 0 . Its explicit expression is given by the formula where K 2 is the modified Bessel function of the third kind and it is a nonnegative function. By the definition of S, see (2.24), we obtain the formula To proceed, we add and subtract to ξ(y ′ ) the function ξ(y) and obtain We claim that − dy dy ′ ξ(y) J ε (y, y ′ ) ξ(y ′ ) − ξ(y) 0. (5.9)

THREE-BODY HAMILTONIAN WITH REGULARIZED ZERO-RANGE INTERACTIONS IN DIMENSION THREE 23
To prove inequality (5.9), we reason as follows. The integral kernel of R λ 0 is (pointwise) positive and χ is a nonnegative function; hence, J ε > 0. Moreover, the expression y) (as one can check with a straightforward calculation), and R λ 0 − 1 2 x + y, − 3 4 x − 1 2 y; x ′ , y ′ shares the same property. Taking also into account the fact that χ is spherically symmetric, it follows that J ε (y, y ′ ) = J ε (y ′ , y). The symmetry of J ε implies dy dy ′ ξ(y) J ε (y, y ′ ) ξ(y ′ ) − ξ(y) from which the inequality (5.9) immediately follows. By (5.8) and (5.9), we find To conclude the proof, we are left to show that (5.10) implies (5.5). The following identity can be obtained by integration starting from identity (5.7) . Hence, To proceed, we use the identity It is easy to check that for k > 0 the function k 0 ds sin s s has maxima in k = nπ for n ∈ N odd, minima in k = nπ for n ∈ N even, it is positive and has an absolute maximum in k = π. Hence, | k 0 ds sin s s | π 0 ds sin s s π. By the latter considerations, we infer Using the latter bound in (5.11) gives dy|ξ(y)| 2 J ε (y) γ 0 dy |ξ(y)| 2 y , (5.12) with the explicit expression for γ 0 . By (5.10) and (5.12) we conclude the proof of the lemma.
Proof. We recall that in the position space the operator Γ reg (see (1.9) and (3.2)) is just the multiplication by the function (denoted by the same symbol) Γ reg (y) = γ y + a(y).
To proceed, we need some further notation. We denote by γ | · | the multiplication operator for γ y and define the operator Similarly to g ε , in the position space the operator ν ε acts as the multiplication by the function (denoted by the same symbol) ν ε (y) = 1 + ε ℓ a(y) Obviously, we have Moreover, there holds the identity 2 for all ε ℓ/(2 a L ∞ ). In the next lemma we study the invertibility of the operator ν * ε Γ λ ε ν ε . Lemma 5.6. For any λ > 0, there holds true the identity where B λ ε is the bounded (although not uniformly in ε) operator Moreover, assume (H1) and γ > γ 0 . Then, ν * ε Γ λ ε ν ε is invertible in L 2 (R 3 ) for all λ > λ 1 and ε small enough, with inverse uniformly bounded in ε and λ.
The last preparatory step is the definition of the bounded operator In Fourier transform Hence (recall that |χ(k)| (2π) −3/2 so that 4π|χ(k)| 2/π) In what follows, we will use the identity We are now ready to formulate and prove the main result of this section.
Proof of Theorem 5.7. The action of H ε on a (symmetric) wavefunction in its domain is given by We describe how to obtain formula (5.21). For a given function φ ∈ L 2 sym (R 6 ), and λ large enough, assume that ψ ε ∈ D(H ε ) ⊂ L 2 sym (R 6 ) is a solution in of the equation The latter gives and, recalling that R λ 0 = (H 0 + λ) −1 is a well defined bounded operator, where we used the fact that R λ 0 and S commute. Hence, Set h ε := χ ε | ⊗ ν * ε ψ ε , and rewrite (5.22) as We want to obtain a formula for h ε . To this aim, apply the operator χ ε | ⊗ ν * ε to (the left of) identity (5.22). By simple algebraic manipulations, it follows that By Lemma 5.6, the operator at the l.h.s. is invertible and where we used the identity A λ * ε = 4π χ ε | ⊗ ν * ε R λ 0 . Using the identity (5.24) in (5.23) we obtain the formula Now, as suggested from the formula above, one can define R λ ε as in (5.21) and show by a straightforward calculation that (H ε + λ)R λ ε = I on L 2 (R 6 ) and R λ ε (H ε + λ) = I on D(H ε ), from which it follows that R λ ε = (H ε + λ) −1 .

Norm Resolvent Convergence
In this section we prove that H ε converges to H in the norm resolvent sense and we give an estimate of the rate of convergence.
which holds true by Lemma 6.2.
Appendix A. Regularity of the Charge In this appendix we characterize D(Γ λ ) and we prove a regularity result for the charge associated with ψ ∈ D(H).
We start by noticing that f λ ∈ L 2 (R 3 ). To see that this is indeed the case recall that by our previous remark ξ ∈ L 2 (R 3 ) and notice that all the terms at the r.h.s. of identity (A.4) are in L 2 . To convince oneself that this is true also for the integral term (for all the others it is obvious), it is sufficient to notice that the integral kernel is an Hilbert-Schmidt operator. To this aim, one can use the inequality and introducing polar coordinates in R 6 verify that the latter integral is finite. To conclude the proof of the proposition we need to show that Decomposingξ andf λ on the basis of Spherical Harmonics, and setting p = e x , ζ ℓm (x) = e 5/2 xξ ℓm (e x ), and h ℓm (x) = e 3/2 xf ℓm (e x ) we obtain: here, clearly,f λ ℓm (p) ∈ L 2 (R + , p 2 dp) for ℓ ∈ N and , m = −ℓ, . . . , ℓ . We look for an inequality between the L 2 -norms of the functions ζ ℓm and h ℓm of the form ζ ℓm c h ℓm with c independent on ℓ and m. To proceed, we decompose (A.3) on the basis of Spherical Harmonics and obtain: =f λ ℓm (p). (A.5) Then, we multiply the latter equation by p 3/2 and change variables as above, with p = e x and q = e y , to obtain: (A.6) The latter equation can be seen as a convolution equation on L 2 (R) and discussed by Fourier transform, to this aim we note the identities: (which hold true because S off,ℓ (k) and S reg,ℓ (k), defined in (3.13) and (3.14), admit an holomorphic extension to the strip {| Im k| < 1}); therefore (see (3.17)) Then, (A.6) is equivalent to To conclude the proof of the proposition, it is sufficient to prove that S ℓ k + i 2 c > 0. We shall focus on the real part of S ℓ and prove that Starting from (3.9) and (3.10), with some straightforward calculations one arrives at cosh k arccos y (−1) n c n y 2 n so that we cannot apply directly Lemma 3.3 to Re S off,ℓ . However, due to the parity properties of the Legendre polynomials, we can apply the lemma for ℓ even and ℓ odd separately. We start with the analysis of the case ℓ odd. We have that where the latter inequality is a consequence of Lemma 3.3.
On the other hand, for ℓ even we have n-even c n y 2 n ℓ -even.
Hence, using again Lemma 3.3, we infer To get the lower bound in the second to last line, we restricted the integral where the second Legendre polynomial is positive, so that we can infer the bound (A.14) by using the monotonicity properties of the integrand. The remaining steps are elementary inequalities. Therefore, by (A.7), together with the lower bounds (A.12), (A.13), and (A.14) we have for ℓ 1 odd Re S ℓ √ 3 2 for ℓ 2 even Re S ℓ √ 3 18 .
(A. 19) We remark that for ℓ = 0 we have a slightly weaker bound involving both f λ and ∆ 1/2 f λ .
By the unitarity of the Fourier transform to prove the bound (A.21) it is enough to find a lower bound for |S ℓ (k + i)| (see also the similar argument used in the proof of Proposition A.2). We concentrate on the real part of S ℓ (k + i).
In the Fourier space equation (A.28) is √