On the Hamiltonian for Three Bosons with Point Interactions

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2 The Thomas paper

In 1935, on G.E. Uhlenbeck’s suggestion, L.H. Thomas investigated the interaction between a neutron and a proton in order to analize the structure of Tritium nucleus [41]. He made the assumption of negligible interaction between the two neutrons and exam- ined how short the neutron-proton interaction could be. He proved that the energy of the system was not bounded below for shorter and shorter interaction range. His con- clusions are clearly stated in the abstract of the paper: “... .We conclude that: either two neutrons repel one another by an amount not negligible compared with the attrac- tion between a neutron and a proton; or that the wave function cannot be symmetrical in their positions; or else that the interaction between a neutron and a proton is not 13 confined within a relative distance very small compared with 10− cm ” . Notice that few years later the connection between spin and statistics was finally clar- ified. It then became clear that the wave function of the two neutrons could not be symmetrical under the exchange of their positions. Nevertheless, Thomas result indi- cated that zero-range interactions, while perfectly defined for a two-particle system, allowing a finite energy for the ground state, seem to give Hamiltonians unbounded from below in the three-boson case. Since then, this effect of “falling to the center” of the quantum three-body system with zero-range interactions is known as Thomas effect (or Thomas collapse). In the paper, Thomas studied, in the center of mass reference frame, the eigenvalue problem for the free Hamiltonian outside a small region around the origin, where the three particles occupy the same position. He found that for negative energy values it was possible to exhibit a square integrable solution showing singularities on the planes where the two neutron positions coincide. The solution was successively used as a test function to show that the energy of the ground state of the system was unbounded from below if the range of the interaction (a sum of two-body potentials or a singular boundary condition) was made shorter and shorter. In particular, let x1, x2 the position of the neutrons and x3 the position of the proton. In the center of mass reference frame let s = x x3 for i = 1, 2 be the relative co- i i − ordinates of the neutrons with respect to the proton. The free Hamiltonian eigenvalue equation then reads

~ 2 (∆s +∆s + s s )Ψ(s1, s2)+ µ Ψ(s1, s2) = 0 µ> 0 . (1) − 4π2m 1 2 ∇ 1 ·∇ 2 Thomas found that a singular solution of (1) is given by

π π 1 2 arctan ξ1 2 arctan ξ2 Ψ(s1, s2)= 2 K0(ηs) − 2 + − 2 (2) s ξ1(1 + ξ ) ξ2(1 + ξ ) 1 2 where

2 2 2 s1 s2 s = s1 + s2 s1 s2; ξ1 = | | ; ξ2 = | | , (3) | | | | − · s1 2s2 s2 2s1 | − | | − | 1 2 4π2m / K0(x) is the zero-th Macdonald function of integer order (see e.g.[19]) and η = ~ µ.

2 We list few important features of the solution. The behaviour of the solution close to the coincidence plane s1 = 0 when s2 > 0 is { } | | 1 π 1 Ψ(s1, s2) K0(η s2 ) . (4) ≈ s1 √3 s2 | | | | | | Symmetrically, when s1 = 0 e s2 > 0 | | ∼ | | 1 π 1 Ψ(s1, s2) K0(η s1 ) . (5) ≈ s2 √3 s1 | | | | | | Moreover, the following scale transformation send eigenvectors in eigenvectors with diﬀerent µ

π π 3 λ 2 arctan ξ1 2 arctan ξ2 Ψλ(s1, s2) λ Ψ(λs1, λs2)= 2 K0(ληs) − 2 + − 2 (6) ≡ s ξ1(1 + ξ ) ξ2(1 + ξ ) 1 2 with λ> 0 and Ψ 2 = Ψ 2. k λk k k Notice that the singularity shown by the solution close to the coincidence planes are the same of the potential of a density charge distributed on the planes. As we will see this behaviour is typical of the functions in the domain of the zero-range interaction Hamiltonians suggested by Skorniakov, Ter-Martirosian [39] and Danilov [20] many years after the work of L. H. Thomas.

3 Zero-range interaction Hamiltonians

It is well known that for system of quantum particles in R3 at low temperature one has that the thermal wavelength

h 2π λ = ~ p ≃ mk T r B is much larger than the range of the two-body interaction. Under these conditions the system exhibits a universal behavior, i.e., relevant observables do not depend on the details of the two-body interaction, but only on a single physical parameter known as the scattering length a := lim f(k, k′) k 0 | |→ where f(k, k′) is the scattering amplitude of the two-body scattering process. In this regime it is natural to expect that the qualitative behavior of the system of particles in R3 is well described by the (formal) eﬀective Hamiltonian with zero-range interactions 1 “H = ∆xi + aijδ(xi xj) ” (7) − 2mi − Xi Xi

3 From the mathematical point of view the first step is to give a rigorous meaning to (7). We define the Hamiltonian for the system of n particles with two-body zero-range interactions as a non trivial self-adjoint (s.a.) extension in L2(R3n) of the operator 1 ∆xi (8) − 2mi Xi defined on H2-functions vanishing on each hyperplane x = x . { i j} The complete characterization of such Hamiltonians can be obtained in the simple case n = 2. Indeed, in the center of mass reference frame, denoting with x the relative coordinate, one has to consider the operator in L2(R3)

1 m1m2 h˜0 = ∆x , m12 = (9) −2m12 m1 + m2 defined on H2-functions vanishing at the origin. Such operator has defect indices (1, 1) and the one-parameter family of s.a. extensions h , α R, can be explicitly constructed. α ∈ Roughly speaking, h acts as the free Hamiltonian outside the origin and ψ D(h ) α ∈ α satisfies the (singular) boundary condition at the origin q ψ(x)= + αq + o(1) for x 0 (10) x | |→ | | where q is a constant depending on ψ and α = a 1. Moreover, the resolvent (h z) 1, − α − − z C R, can be explicitly computed and all the spectral properties of h characterized ∈ \ α (see [2], [5]). For systems of n particles, with n> 2, the construction is much more difficult because of the presence of infinite dimensional defect spaces (see, e.g., [3], [4], [6], [8], [9], [15], [16], [17], [18], [22], [23], [25], [26], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [40]). To simplify notation, we describe the problem, and some previous attempts to solve it, in the case of three identical bosons with masses 1/2, in the center of mass reference frame. Let x1, x2 and x3 = x1 x2 be the cartesian coordinates of the particles. Let us − − introduce the Jacobi coordinates 1 x = x2 x3 , y = (x2 + x3) x1 (11) − 2 − 2 1 1 1 1 with inverse given by x1 = y , x2 = x + y , x3 = x + y . Due to the − 3 2 3 − 2 3 symmetry constraint, the Hilbert space of states is 1 3 1 L2(R6)= ψ L2(R6) s.t. ψ(x, y)= ψ( x, y)= ψ x + y, x y (12) s ∈ − 2 4 − 2 n o and the formal Hamiltonian reads 3 “H = ∆x ∆y + aδ(x)+ aδ(y x/2) + aδ(y + x/2) ”, (13) − − 4 − i.e., H is a perturbation of the free dynamics in R6 supported by the three-dimensional hyperplanes Σ= x = 0 y x/2 = 0 y + x/2 = 0 . (14) { } ∪ { − } ∪ { } 2 R6 According to our mathematical definition, a s.a. Hamiltonian in Ls( ) corresponding to the formal operator H is a non trivial s.a. extension of the operator

3 2 6 2 6 H˜0 = ∆x ∆y , D(H˜0)= ψ L (R ) s.t. ψ H (R ), ψ = 0 . (15) − − 4 ∈ s ∈ Σ n o

4 As we already mentioned the defect indices are now infinite and the problem arises of how to choose and characterize a class of s.a. extensions with the right physical properties. An apparently reasonable choice based on the analogy with the case n = 2 is due to Ter-Martirosian and Skorniakov [39]. Indeed, they defined an operator Hα acting as the free Hamiltonian outside the hyperplanes and satisfying a boundary condition at the hyperplanes. Specifically, they impose

ξ(y) ψ(x, y)= + αξ(y)+ o(1) , for x 0 and y = 0 (16) x | |→ 6 | | where ξ is a function depending on ψ. Due to the bosonic symmetry, the same conditions for y x/2 0 and y + x/2 0 have to be satisfied. | − |→ | |→ We note that the first term in the right-hand side of (16) coincides with the first term of the asymptotic expansion of the potential produced by the charge density ξ distributed on x = 0 . Therefore, an equivalent way to describe a wave function ψ in the domain { } of H is the following. Any ψ D(H ) can be decomposed as α ∈ α ψ = wλ + λξ , wλ H2(R6) (17) G ∈ where λ> 0 and

2 ξˆ(p)+ ξˆ(k 1 p)+ ξˆ( k 1 p) λξ(k, p)= − 2 − − 2 . (18) G π k 2 + 3 p 2 + λ r | | 4 | | d Note that the function λξ(x, y) has the asymptotic behaviour G

λ ξ(y) 1 ip y λ ξ(x, y)= dp e · T ξˆ (p)+ o(1) (19) G x − (2π)3/2 | | Z for x 0 and y = 0, where | |→ 6 ˆ λ 3 2 1 ξ(p′) T ξˆ (p) := p +λ ξˆ(p) dp′ . (20) 4| | − π2 p 2 + p 2 + p p + λ r Z | | | ′| · ′ We will refer to the ﬁrst and second term in (20) respectively as the diagonal and the non-diagonal part of T λ. Therefore the boundary condition (16) is now rewritten as

ˆ ˆ λ α ξ(p)+ T ξ (p) = (w x=0)∧(p) . (21) There is an ambiguity in the above definition since the domain of the symmetric and unbounded operator T λ in L2(R3) is not specified. As a first attempt one can choose

D(T λ)= ξˆ L2(R3) dk k 2 ξˆ(k) 2 < ξˆ ξ H1(R3) . (22) { ∈ | | | | | ∞}≡{ | ∈ } Z Note that for ξˆ D(T λ) both terms in the r.h.s. of (20) belong to L2(R3) (see, e.g., ∈ [23]). As a matter of fact, the operator Hα deﬁned in this way is symmetric but not s.a. and it turns out that its s.a. extensions are all unbounded from below. This fact, ﬁrst noted by Danilov [20], was rigorously analyzed by Minlos and Faddeev in [34], [35].

5 4 Minlos and Faddeev contributions

In the first of their seminal contributions (see [34]) Minlos and Faddeev consider a system of three bosons and approach the general mathematical problem to give a meaning to the formal Hamiltonian with zero-range interactions. Working in momentum space and using the theory of s.a. extensions of semibounded operators developed by Birman [10], they obtain the abstract characterization of all the s.a. extensions of the operator (15). 2 R6 Then they observe that the s.a. extensions of the operator Hα in Ls( ) introduced by Ter-Martirosian and Skornyakov are in one-to-one correspondence with the s.a. extensions of the operator T λ in L2(R3). Such operator T λ defined on D(T λ) has defect indices (1, 1) and they find that a s.a. extension T λ, β R, of T λ is defined on β ∈

λ 2 3 λ D(T )= ξˆ L (R ) ξˆ = ξˆ1 + ξˆ2, ξˆ1 D(T ), β ∈ | ∈ n c ξˆ2(k)= β sin s0 log k + cos s0 log k (23) k 2 + 1 | | | | | | o where c is an arbitrary constant and s0 is the positive solution of the equation

πs 8 sinh 6 1 πs = 0 . (24) − √3 s cosh 2 One can observe that both the diagonal and the non diagonal parts of T λ diverge on functions with the asymptotic behaviour of ξˆ2 in (23) for k , but their sum | |→∞ remains ﬁnite. λ λ Given the s.a. operator Tβ , D(Tβ ), one obtains the s.a. extension Hα,β (also called Ter-Martirosian, Skornyakov Hamiltonian) of Hα

D(H )= ψ L2(R6) ψ = wλ + λξ, wλ H2(R6), ξˆ D(T λ), α,β ∈ s | G ∈ ∈ β n ˆ λ ˆ λ α ξ(p)+ T ξ (p) = (w x=0)∧(p) , (25) λ (H + λ)ψ = (H0 + λ)w , o (26) α,β where 3 2 6 H0 = ∆x ∆y , D(H0)= H (R ) . (27) − − 4 Roughly speaking, β parametrizes a further boundary condition satisfied at the triple coincidence point x1 = x2 = x3 = 0. Therefore, it can be considered as the strength of a sort of an additional three-body force acting on the particles when all their positions coincide. The authors conclude claiming that some further results on the spectrum of the Hamil- tonian Hα,β hold. In particular, they affirm that Hα,β has the unphysical instability property already noted by Danilov, i.e., that there exists an infinite sequence of negative eigenvalues E , as n . n → −∞ →∞ The rigorous proof of this fact is contained in their second paper on the subject and it will be described below. At the end of the paper one finds an interesting remark on the possibility to define a modified Hamiltonian satisfying the stability property, i.e., bounded from below. The authors say: “ We note that this last result (i.e., the instability property) somewhat discredits our chosen extension, since probably only semibounded energy operators are

6 of interest in nonrelativistic quantum mechanics. It seems to us that there must exist among the other extensions of the operator H˜0 semibounded extensions which have all the properties of the model of Ter-Martirosian and Skornyakov that are good from the physical point of view, namely the properties of locality and of the correct character of the continuous spectrum. Evidently such extensions will be obtained ... ” if one replaces the constant α in (21) (or equivalently in (16)) with the operator αM in the Fourier space deﬁned by

(αM ξˆ)(p)= αξˆ(p) + (Kξˆ)(p) (28) where α R and K is a convolution operator with a kernel K(p p ) having the ∈ − ′ asymptotic behavior γ K(p) , for p (29) ∼ p 2 | |→∞ | | with the constant γ satisfying 1 4π γ > 1 . (30) π3 3√3 − Unfortunately, they conclude: “A detailed development of this point of view is not presented here because of lack of space.” We believe that such suggestion is interesting and we find it rather strange that the idea has never been developed in the literature. We also observe that it is not so evident that the replacement of the constant α with the operator αM defined in (28), (29), (30) produces a semibounded Hamiltonian and, moreover, it is not clear the physical meaning of such replacement. We shall come back to this point in the next section. Here we continue the analysis of the contribution of Minlos and Faddeev discussing the content of their second paper on the subject ([35]), where the authors show that the Hamiltonian Hα,β has an infinite number of eigenvalues accumulating both at zero and at . We give here a slightly different, and more elementary, proof than the one given −∞ in [35]. To simplify the notation, we consider only the case α = 0. We recall that α = 0 corresponds to a two-body interaction with zero-energy resonance (see, e.g., [2]). Taking into account definitions (25), (26), an eigenvector of H0,β associated to the negative eigenvalue E = µ, µ> 0, has the form µξ, where ξˆ D(T ) is a solution of − G ∈ β the equation

ˆ 3 2 1 ξ(p′) p +µ ξˆ(p) dp′ = 0 . (31) 4| | − π2 p 2 + p 2 + p p + µ r Z | | | ′| · ′ We shall compute the rotationally invariant solutions ξˆ = ξˆ( p ) of (31). Performing | | the angular integration in the second term of (31), one obtains the equation

2 2 3 2 ˆ 2 ∞ ˆ p + p′ + pp′ + µ p +µ p ξ(p) dp′ p′ξ(p′) log 2 2 = 0 . (32) 4 − π 0 p + p pp + µ r Z ′ − ′

Let us introduce a change of the independent variable

2√µ √3p 3p2 p = sinh x, x = log + + 1 (33) √3 2√µ s 4µ !

7 and deﬁne 2 µ sinh x cosh x ξˆ √µ sinh x for x 0 θ(x)= √3 ≥ (34) ( θ( x) for x< 0 − − so that √3p 3p2 2 θ log 2√µ + 4µ + 1 ξˆ(p)= . (35) √3 h 3 4q i p 4 p + µ The ﬁrst term in (32) in the new coordinatesq is

3 2√µ 2√µ p2 +µ p ξˆ(p)= µ sinh2x + µ sinh x ξˆ sinh x 4 √3 √3 r q 2 = θ(x) . (36) √3 The other term in (32) in the new coordinates reads

2 2 2 ∞ ˆ p + p′ + pp′ + µ dp′ p′ξ(p′) log 2 2 − π 0 p + p′ pp′ + µ Z 2 − 2 8 ∞ sinh x + sinh y + sinh x sinh y + 3/4 = dy θ(y) log 2 2 −3π 0 sinh x + sinh y sinh x sinh y + 3/4 Z − 8 ∞ (2 cosh(x + y) 1) (2 cosh(x y) + 1) = dy θ(y) log − − −3π 0 (2 cosh(x + y) + 1) (2 cosh(x y) 1) Z + − − 8 ∞ 2 cosh(x y) + 1 = dy θ(y) log − (37) −3π 2 cosh(x y) 1 Z−∞ − − where in the last line we have used the extension θ(x)= θ( x) for x< 0. Therefore, − − equation (32) for ξˆ(p) is transformed into the following convolution equation for θ(x)

+ 4 ∞ 2 cosh(x y) + 1 θ(x) dy θ(y) log − = 0 . (38) − √3π 2 cosh(x y) 1 Z−∞ − − Finally, we compute the Fourier transform (see [19], p. 36) and we arrive at the equation for θˆ

8 sinh π s 1 6 θˆ(s) = 0 . (39) − √3 s cosh π s 2 Denote by g(s) the function in parenthesis in (39). It is easy to see that g is even, monotone increasing for s> 0 and g(s) 1 for s + . Moreover, g(0) = 1 4π < 0 → → ∞ − 3√3 and we conclude that the equation g(s) = 0 has two solutions s = s0, with s0 > 0. ± Since θˆ is an odd function, the solution of (39) reads

θˆ(s)= δ(s s0) δ(s + s0) (40) − − apart from a multiplicative constant and therefore

θ(x) = sin s0x . (41) From (35) we obtain the solution of equation (32) for any µ> 0

8 √3p 1 3 2 sin s0 log 2√µ + √µ 4 p + µ ξˆµ(p)= . (42) h 3 2 q i p 4 p + µ q We note that the solution (42) belongs to L2(R3) but it does not belong to D(T λ) 2 because of the behavior O(p− ) for p . On the other hand we can ﬁnd suitable → ∞ λ values of µ such that the solution belongs to D(Tβ ). 4µ 2 Indeed, denoting ε = 3 p− , we have

√3p ε/2 sin s0 log + log 1+ √µ 1+√1+ε ξˆµ(p)= n h p2 √1+ε io

s0 3 sin s0 log p + 2 log µ = + η1(p) p2√1+ ε s0 3 sin (s0 log p) s0 3 cos (s0 log p) = cos log + sin log + η2(p) (43) 2 µ p2 + 1 2 µ p2 + 1 λ λ where η1, η2 D(T ). According to (23), in order to have ξˆ D(T ) we impose the ∈ µ ∈ β condition

s0 3 s0 3 cos log = β sin log . (44) 2 µ 2 µ Condition (44) is satisﬁed if and only if µ is equal to

π 2 cot−1 β 2 n µ = 3 e− s0 e s0 , n Z . (45) n ∈ Thus we obtain an inﬁnite sequence of negative eigenvalues

E = µ , n Z (46) n − n ∈ µn with corresponding eigenvectors ξ n , where ξˆ n is given by (42). G µ µ We stress that the model Hamiltonian H0,β exhibits the Efimov effect, i.e., there exists an infinite sequence of eigenvalues E 0 for n satisfying the (exact) n → → −∞ geometrical law π En 2 = e− s0 . (47) En+1 On the other hand, one also has E for n + , corresponding to the instability n → −∞ → ∞ property known as Thomas effect.

5 Regularized zero-range interactions

In this section we propose a model Hamiltonian for three bosons with zero-range interactions regularized around the triple coincidence point x1 = x2 = x3 = 0 in such a way to avoid the Thomas effect, i.e., with a spectrum bounded from below. We follow the proposal contained in [4] which, in turn, essentially coincides with the already mentioned suggestion discussed at the end of [34]. In the first part of [4] the authors announce an interesting mathematical result on the Efimov effect. They consider a three-particle system with Hamiltonian H, with

9 two-body, spherically symmetric, short range potentials such that at least two of the two-body subsystems have zero-energy resonances (i.e. inﬁnite scattering length). They claim that H has inﬁnitely many spherically symmetric bound states with energy E n → 0 such that En 2π lim = e− σ (48) n +1 →∞ En where σ > 0 depends only on the mass ratios (and coincides with s0 in (47) in the case of three identical bosons). The result should follow “from a detailed study of the asymptotic behavior of the action of the scaling group in the spaces of two and three- body Hamiltonians restricted to functions invariant under the natural action of SO(3)”, where the scaling group is given by

1 1 1 x H Hε = UεHU − , (Uεψ)(x)= ψ , ε> 0 . (49) → ε2 ε ε3/2 ε Roughly speaking, for ε 0 the rescaled Hamiltonian should converge to the zero-range → En 2π/σ model H0 . Since for H0 the Efimov effect, together with the property = e , ,β ,β En+1 − is explicitly verified, one should infer the result for H. Unfortunately, this program has not been realized and it remains as a challenging open problem. In the last part of the paper the authors add an interesting remark for our aim concerning the construction of a reasonable three-body Hamiltonian with zero-range interactions which is bounded from below. Indeed, they claim that one can consider a Hamiltonian with zero-range interactions where the zero-range force between two particles depends on the position of the third one. If such three-body force is suitably chosen then the Hamiltonian is bounded from below. The proposed recipe can be rephrased in the following way: in the boundary condition (16) one replaces the constant α with a position-dependent term

δ α (y)= α + , α, δ R . (50) A y ∈ | | In the case of equal masses, they affirm that for 2 4π δ > 1 (51) π2 3√3 − the corresponding zero-range Hamiltonian is bounded from below. Also the proof of this statement is postponed to a forthcoming paper which has never been published. Let us briefly comment on the above proposal. The replacement of the constant α with a function α(y), with α(y) for y 0, has a reasonable physical meaning. It → ∞ | | → means that when the positions of the three particles coincide, i.e., for x = y = 0, the two-body interactions are switched off (α = means no interaction). In this way one ∞ compensates the tendency of the three interacting particles to “fall in the center”. On the other hand, the specific choice of the function αA(y) is not explained in the paper but one can imagine that such a function allows some explicit computations (as in the case of the choice (29) of the operator K in [34]). It is also natural to compare the two proposals contained in [34] and [4]. It is immediate to realize that the two proposals essentially coincide in the sense that one is the Fourier transform of the other, i.e.,

2 (αAξ)∧(p) = (αM ξˆ)(p) , if δ = 2π γ . (52)

10 It is also important to stress that only the asymptotic behavior of K(p) for p (see | |→∞ (29)) is relevant to obtain a lower bounded Hamiltonian, as it is correctly pointed out in [34]. Correspondingly, it must be suﬃcient to require only the asymptotic behavior δ y 1 + O(1) for y 0 for the position-dependent strength of the interaction. | |− | |→ In conclusion, following the (common) idea proposed in [34] and [4], we introduce the Hamiltonian Hα˜ characterized by the boundary condition ξ(y) ψ(x, y)= +α ˜( y ) ξ(y)+ o(1) , for x 0 and y = 0 (53) x | | | |→ 6 | | where δ ℓ α˜ : R+ R , α˜(r)= α + χ (r) (54) → r with 1 r ℓ α R, δ> 0, ℓ> 0, χℓ(r)= ≤ (55) ∈ ( 0 r>ℓ. More precisely, we deﬁne the Hamiltonian as follows

2 6 λ λ λ 2 6 1 3 D(H˜)= ψ L (R ) ψ = w + ξ, w H (R ), ξ H (R ), α ∈ s | G ∈ ∈ n λ ˆ λ ∧ (˜αξ)∧(p)+ T ξ (p) = w x=0 (p) , (56) λ (H˜ + λ)ψ = (H0 + λ)w . o (57) α In a paper in preparation ([7]) it will be proved the following result.

Proposition 5.1. Let δ > δ0, where √3 4π δ0 = 1 . (58) π √ − 3 3 Then for any ℓ (0, + ] and α R the operator (56), (57) is s.a. and bounded from below. ∈ ∞ ∈ The result shows that it is suﬃcient to add a three-body force with an arbitrary small (but diﬀerent from zero) range to avoid the collapse. We stress that it would be interesting to prove that boundedness from below is preserved taking “some suitable limit ℓ 0”. → The proof is based on the analysis of the quadratic form associated to Hα˜. Taking into account of (56), (57), by an explicit computation for ψ D(H˜) one obtains ∈ α λ λ λ λ λ (ψ, (H˜ + λ)ψ) = (ψ, (H0 + λ)w )=(w , (H0 + λ)w )+( ξ, (H0 + λ)w ) α G λ λ λ = (w , (H0 + λ)w ) + 12π (ξ, αξ˜ ) + (ξ,ˆ T ξˆ) . (59)

Hence we deﬁne the quadratic form h i λ λ 2 λ F˜(ψ) = (w , (H0 + λ)w ) λ ψ + 12 π Φ (ξ) (60) α − k k α˜ where λ 2 λ Φ˜(ξ)= dy α˜( y ) ξ(y) + dp ξˆ(p)(T ξˆ)(p) (61) α | | | | Z Z and 2 6 λ λ λ 1 6 1/2 3 D(F˜)= ψ L (R ) ψ = w + ξ, w H (R ), ξˆ H (R ) . (62) α ∈ s | G ∈ ∈ The proof proceedsn taking such a quadratic form as starting point and provingo that it is closed and bounded from below. Therefore it uniquely deﬁnes a s.a. and bounded from below operator which coincides with (56), (57).

11 6 On the negative eigenvalues

In this section we consider the eigenvalue problem for Hα˜ in the special case

α = 0, ℓ =+ . (63) ∞ As we already noticed in the case of the TMS Hamiltonian, an eigenvector associated to the negative eigenvalue E = µ, µ > 0, has the form µξ, where ξ H1(R3) is a − G ∈ solution of the equation

δ ∞ p + p′ 3 2 dp′ p′ξˆ(p′) log + p +µ p ξˆ(p) π 0 p p′ r4 Z | − | 2 2 2 ∞ ˆ p + p′ + pp′ + µ dp′ p′ξ(p′) log 2 2 = 0 . (65) − π 0 p + p pp + µ Z ′ − ′ In the following we prove that for δ > δ0 there are no solutions of (65) and therefore the Hamiltonian has no negative eigenvalues corresponding to rotationally invariant solutions of (64). The main point of the proof is that the l.h.s. of (65) can be diagonalized by the same change of coordinates used in section 2 for the TMS Hamiltonian.

Proposition 6.1. Let δ > δ0. Then equation (65) has only the trivial solution.

Proof. Using (33), (34), for the ﬁrst term in (65) we have

δ ∞ p + p′ 4δ ∞ sinh x + sinh y dp′ p′ξˆ(p′) log = dy θ(y) log π 0 p p 3π 0 sinh x sinh y Z | − ′| Z − x+y x y 4δ ∞ sinh 2 cosh −2 = dy θ(y) log x+y x y 3π 0 cosh 2 sinh −2 Z x y x+y 4δ ∞ cosh −2 4δ ∞ sinh 2 = dy θ(y) log x y + dy θ(y) log x+y 3π 0 sinh − 3π 0 cosh Z 2 Z 2 + 4δ ∞ x y = dy θ(y) log coth − (66) 3π 2 Z −∞ where in the last line we have used the extension θ(x)= θ( x) for x< 0. Hence, by − − (36), (37) and (66), in the new variables equation (65) reads

+ 2δ ∞ x y θ(x)+ dy θ(y) log coth − √3π 2 Z −∞ + 4 ∞ 2 cosh(x y) + 1 dy θ(y) log − = 0 . (67) − √3π 2 cosh(x y) 1 Z−∞ − −

12 We note that

1 isx x 2 ∞ x dx e− log coth = dx cos sx log coth √2π 2 π 0 2 Z r Z 2 ∞ x 2 ∞ x = dx cos sx log 1+ e− dx cos sx log 1 e− . (68) π − π − r Z0 r Z0 Then we use [24], page 582, and (39) to write the equation for the Fourier transform of θ δ sinh π s 4sinh π s 2 − 6 ˆ 1 + 2 π θ(s) = 0 . (69) √3 s cosh 2 s !

It remains to show that the function in parenthesis in (69) is strictly positive for δ > δ0. Note that the function is even and then we can consider s 0. For δ > δ0 it is positive ≥ in s = 0. Moreover, if we denote π π π F (s)= √3 s cosh s + 2δ sinh s 8sinh s , (70) 2 2 − 6 for δ > δ0 we have

π √3 π π 4π π F ′(s) = (√3+ πδ) cosh s + s sinh s cosh s 2 2 2 − 3 6 π 4π π > (√3+ πδ0) cosh s cosh s 2 − 3 6 4π π √3+ πδ0 cosh s = 0 (71) ≥ − 3 6 so that F (s) > 0 for any s. It follows that equation (69), and then equation (65), has only the trivial solution and this conclude the proof of the proposition.

7 Acknowledgements by way of conclusion

We have many reasons for being grateful to Sergio Albeverio, mainly for his lasting friendship and for all the affection and help he has been giving us for so many years. Among all these good reasons, one concerns the support he provided to our scientific activity. As it is very well known, he was one of the main players in the scientific project aimed to explicitly construct models in relativistic quantum field theory describing interacting bosons, a project that never reached a final result in four dimensional space-time. The formal non-relativistic limits of the relativistic models under study in constructive quantum field theory (see [21] for the only rigorous result in the investigation of such limits) describe particles in three dimensions interacting via zero-range forces. This was the reason why, together with many other scientific interests, he got involved in the theory of point interaction Hamiltonians. In the early nineties he suggested to one of us (A.T., young postdoc in Bochum at that time) to investigate further the spectral structure of the point interaction Hamiltonian for a three-boson system, following the suggestions given in [4]. With some delay, together with a group of other younger italian researchers, we figured out better those suggestions and started working in this direction. In this contribution we outlined the history of the main achievements in this research field and we sketched our first results.

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